# Using a Local Surrogate Model to Interpret Temporal Shifts in Global Annual
Data

Shou Nakano  Wilfrid Laurier Univeristy
Waterloo, Canada
<EMAIL_ADDRESS>Yang Liu  Wilfrid Laurier University
Waterloo, Canada
<EMAIL_ADDRESS>
###### Abstract

This paper focuses on explaining changes over time in globally-sourced, annual
temporal data, with the specific objective of identifying pivotal factors that
contribute to these temporal shifts. Leveraging such analytical frameworks can
yield transformative impacts, including the informed refinement of public
policy and the identification of key drivers affecting a country’s economic
evolution. We employ Local Interpretable Model-agnostic Explanations (LIME)
[1] to shed light on national happiness indices, economic freedom, and
population metrics, spanning variable time frames. Acknowledging the presence
of missing values, we employ three imputation approaches to generate robust
multivariate time-series datasets apt for LIME’s input requirements. Our
methodology’s efficacy is substantiated through a series of empirical
evaluations involving multiple datasets. These evaluations include comparative
analyses against random feature selection, correlation with real-world events
as elucidated by LIME, and validation through Individual Conditional
Expectation (ICE) [2] plots, a state-of-the-art technique proficient in
feature importance detection.

###### Index Terms:

XAI, Interpretable Machine Learning, LIME

##  I Introduction

This paper investigates temporal data collected from global sources on an
annual basis. Our primary focus lies in the effective utilization of advanced
data analytics to forecast emerging trends and identify avenues for
improvements at the national level. The insights gained from these analyses
have far-reaching implications, including the potential to guide policy
reforms that could significantly enhance the well-being of citizens.
Additionally, our work offers valuable perspectives on the key factors that
play a pivotal role in shaping a country’s economic direction, among other
applications.

LIME has traditionally been employed to dissect and interpret model
predictions. Its adaptability has been showcased in areas like time-series
data analysis [3] and the interpretation of cardiographic signals [4].
Furthermore, it has been pivotal in formulating explanations that precisely
illuminate a model’s workings. A significant innovation in this study is the
extension of LIME’s scope to encompass datasets featuring multiple time-
series, marking a departure from the customary singular time-series modality.
This paradigm shift is paramount for dissecting annual changes across nations.
To facilitate this, we propose a list of approaches to handle missing values
and reform datasets of different formats.

In addition, LIME is deployed to scrutinize predictions generated by models
fine-tuned using FLAML [5], a Python library adept at autonomously pinpointing
optimal hyperparameters across a spectrum of intricate models. By delving into
an assortment of black-box models, this research aims to elucidate LIME’s
proficiency in distilling these models and elucidating the foundational
rationale of their predictions. Furthermore, it offers insights into how
alterations in features influence the predicted outcomes, enhancing user
comprehension.

This study employs three open-access time-series datasets that capture
variations in national happiness levels, economic freedom, and population
dynamics. We assess the validity of our approach through a series of tests
that include: benchmarking against randomly selected features, establishing
correlations with real-world occurrences as interpreted by LIME, and cross-
validating outcomes using Individual Conditional Expectation (ICE) plots, a
leading method for determining feature significance. The focus lies on the
precision of LIME in feature identification, supported by concrete case
studies that authenticate its predictive claims, regardless of whether they
carry positive or negative implications.

##  II Related Work

Introduced in 2016, LIME aimed to understand black box models by generating
interpretable and accurate predictions via the creation of a more transparent
model [1]. This transparency is achieved by iteratively sampling the intricate
model to approximate its behavior. Given LIME’s approach of treating the model
as a black-box, it exhibits model-agnostic properties. Then, from LIME’s
recreation of the model, individual points can be provided for LIME to provide
explanations for. This approach was expanded upon by several researchers who
noted that LIME lacked any support for time-series data with thousands of
entries, such as hourly weather updates or stock prices, leading to the
proposal of LEFTIST, the first model-agnostic approach for time series
data[3], which segmented the time series data into several components of equal
length. This was followed by TS-MULE [6], which approached several problems
like how to properly segment the time series into several windows of different
lengths, such as SAX segmentation and suggested several transformations.
Finally, LIMESegment was developed a year later [7], which outperformed
previous research with an approach involving frequency distributions and
understanding the local neighborhood of a time series.

Previous work has also been done in terms of applying LIME to various
datasets, since LIME is highly successful and can be easily applied to
textual, tabular and image data. For example, LIME has been used to categorize
species of plants after InceptionV3, XCeption and ResNet50 models were trained
[8], to show why image captioning models selected certain words to be a part
of an image’s caption [9], to classify healthy and unhealthy hair through
pictures of various scalps in order to detect diseases like folliculitis
decalvans and acne keloidalis, with a high accuracy of 96.63% from the black
box model that was backed up with explainable machine learning.

Several other applications exist, such as using LIME in conjunction with other
approaches like Shapley Values to explain why users could potentially have
diseases like diabetes and breast cancer [10]. LIME has also been used to
provide real-world explanations for why a synthetic dataset of various machine
parts would suffer breakdowns, with explanations being provided based on the
parts’ wear and tear, as well as the temperatures that they were exposed to
[11]. From the observations, it’s evident that LIME can efficiently generate
insightful explanations for a model, provided the model is precise and the
data type is compatible with LIME or its variants like LIMESegment. However,
for the scope of this study, LIMESegment and other approaches to univariate
time-series are not valid choices, since LIMESegment works with univariate
time series, not multivariate ones. Thus, an algorithm like LIMESegment will
only be able to look at the year column and the target value, instead of
combining them with other features to glean more information. Therefore, a
strategy that encapsulates the entirety of the data and works with any tabular
dataset, not just a specific group of datasets is essential, ensuring the
model has an optimal amount of data for processing.

##  III Methodology

###  III-A Problem Definition

In this study, our objectives are twofold. Firstly, given an input
$X=(x_{1},x_{2},\ldots,x_{n})$, we aim to predict the target label $y$ with
the highest accuracy utilizing sophisticated models. The data at hand might
necessitate imputation and is bound to undergo adjustments for temporal
scrutiny. The adoption of intricate models serves to demonstrate LIME’s
capability to render a model more comprehensible irrespective of its inherent
complexity. Formally, LIME can be mathematically represented as Equation (1).
Here, $L$ denotes the loss, capturing the discrepancy between LIME’s
approximated model, in this case, a linear model $g$ and the actual model $f$.
$\pi_{x}$ delineates the extent of the vicinity around $x^{\prime}$, the point
that was passed into LIME, where the vicinity is where points will be sampled,
while $\Omega(g)$ signifies the model’s complexity. Both $L$ and $\Omega(g)$
should be minimized.

$explanation(x)=\underset{g\in G}{\arg\min}L(f,g,\pi_{x})+\Omega(g)$ (1)

Secondly, our aim is to showcase the precision and capabilities of LIME when
applied to the adapted data. This is achieved by comparing LIME’s outcomes and
predictions with real-world instances, and by contrasting the Individual
Conditional Expectation Plots with LIME’s adaptation of the Submodular Pick
[2]. The latter addresses an NP-Hard challenge, where the objective is to
generate explanations for points, ensuring maximal distinctiveness. The
underlying mathematics for the submodular pick algorithm can be shown through
Equation (2) and Equation (3).

$c(V,W,I)=\sum_{j=1}^{d^{\prime}}\mathbbm{1}_{[\exists i\in
V:W_{i,j}>0]}I_{j}$ (2)

$Pick(W,I)=\underset{V,|V|\leqslant B}{\arg\max}c(V,W,I)$ (3)

Equation (2) offers a definition of coverage by examining $V$, a collection of
instances, alongside $W$, representing the local significance of each
instance. $I_{j}$ symbolizes the global prominence of a specific element
within the explanation domain. The formulation of $I_{j}$ is contingent upon
the data type and can be refined for enhanced efficiency. Equation (3) focuses
on the pick problem, striving to amplify the cumulative coverage achieved by
selecting multiple local instances. This is realized by leveraging the values
computed from Equation (2).

###  III-B Data Collection

Three datasets were collected for the purpose of this experiment, where the
first originates from the World Happiness Report [12], the second dataset was
collected from Kaggle [13], which was initially sourced from the Fraser
Institute and the third dataset comes from the U.S.A.’s Census Bureau [14].
Each dataset is predominantly numerical, aside from the names of countries.
Additionally, the values in the first two datasets range from 0 to 10.

Dataset 1: The World Happiness Report dataset [12] compiles yearly data from
2015 to 2022 by gathering polls from thousands of citizens to understand how
they feel about their country and these results are then used to predict a
happiness score. Over the years, the data collection methods, naming
conventions, and columns have evolved. The final dataset retained columns such
as Country Name, Year, Life Ladder, GDP per Capita, Social Support, Healthy
Life Expectancy, Freedom to Make Life Choices, Generosity, Perception of
Corruption, and Dystopia + Residual. Country names that had changed from 2015
to 2022, like Swaziland to Eswatini, were manually updated. This dataset was
chosen to investigate if any particular features would greatly affect a
country’s happiness score.

Dataset 2: The second dataset was originally published by the Fraser Institute
and later reformatted on Kaggle [13], focuses on the economic freedom of
various countries. It is organized into five main categories, each containing
multiple sub-categories, totaling 25 factors. Each country’s economic freedom
score lies between 0 and 10. Such categories include military interference,
inflation, business regulations, etc. The dataset’s linear relationships allow
for a relatively easy calculation of a country’s economic freedom, yet it also
serves as a useful tool for case studies and evaluating the effectiveness of
LIME with FLAML-trained models.

Dataset 3: The third dataset is a comprehensive set of data published by the
U.S.A. [14], which tracks countries’ population year by year and provides a
detailed background of its’ sex ratio, fertility rate, life expectancy,
mortality rate, crude death rate, among others. This dataset provides
hypothetical estimates of the population up to 2100, but for the sake of
realism in this report, the dataset has been truncated at 2023. This dataset
aims to evaluate how real-world, non-calculated Y-column data performs in
temporal analyses and with the LIME approach.

###  III-C Preprocessing

Addressing Missing Data: Managing data with missing values is a pivotal issue
in both data analytics and model building. This concern is especially
pronounced in our investigation involving the second dataset, where nearly
$65\%$ of the entries had at least one column with missing data. To tackle
this challenge, we utilized three distinct imputation methods, ultimately
selecting the best-performing approach for data imputation. This process is
illuatrated with Algorithm 1.

  1. 1.

Linear Regression Imputation: To predict missing values, we treat the feature
with missing data as the dependent variable and use the other features which
are complete (or mostly complete) as the independent variables. Formally, if a
value $y_{i}$ is missing, its imputed counterpart $\hat{y}_{i}$ can be
calculated using the equation:

$\hat{y}_{i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{i1}+\hat{\beta}_{2}x_{i2}+\ldots+\hat{\beta}_{n}x_{in}$

Here, $\hat{\beta}_{0},\hat{\beta}_{1},\ldots,\hat{\beta}_{n}$ represent the
estimated coefficients obtained from the linear regression model trained on
the observed data. The imputed value $\hat{y}_{i}$ then takes the place of the
missing value. Before using the model for imputation, we adopt Ordinary Least
Squares (OLS), which minimizes the sum of squared differences between observed
and predicted values, to estimate $\beta$. This approach intuitively leverages
relationships between variables to estimate absent data, ensuring that the
inherent correlations in the dataset are considered.

  2. 2.

KNN Imputation: We estimate a missing value by considering its ‘k’ most
similar data points (neighbors) that have no missing values for that specific
feature. The task is further divided into two steps:

    1. (a)

Finding Neighbors: We identify the $k$ nearest neighbors based on Euclidean
distance metric

$d(x,y)=\sqrt{\sum_{i=1}^{n}(x_{i}-y_{i})^{2}}.$

    2. (b)

Computing the Imputed Value: The imputed value $x_{\text{imputed}}$ is then
calculated as the weighted average of these $k$ neighbors:

$x_{\text{imputed}}=\frac{\sum_{i=1}^{k}w_{i}x_{i}}{\sum_{i=1}^{k}w_{i}}$

where $w_{i}$ are the weights, corresponding to the inverse of the distance to
the neighbors.

This method offers the advantage of flexibility, as it is suitable for both
numerical and categorical data and does not rely on linear assumptions,
allowing it to handle non-linear relationships in datasets effectively.
However, the method can be computationally intensive, especially for large
datasets, due to the need to calculate distances and identify nearest
neighbors.

  3. 3.

Iterative Imputation: We take a stepwise approach to model each feature with
missing values as a function of other features in an iterative manner. The
process begins with an initial imputation with mean values. Each feature with
missing values is then treated as a dependent variable, with the remaining
features acting as independent variables. The missing values are predicted
iteratively, cycling through each feature, until the imputations stabilize.
Mathematically, the imputed value for a missing entry $x_{ij}$ at iteration
$t$ is given by
$x_{ij}^{(t)}=\beta_{0}^{(t)}+\beta_{1}^{(t)}x_{1j}+\ldots+\beta_{n}^{(t)}x_{nj}+\epsilon_{j}^{(t)}$,
where $\beta_{0}^{(t)},\beta_{1}^{(t)},\ldots,\beta_{n}^{(t)}$ are the
estimated coefficients and $\epsilon_{j}^{(t)}$ is a random residual.

Data: Two-dimensional dataset D with missing values, threshold $\theta$ for
missing rate

Result: Imputed dataset

Initialize an empty dataset imputed_data;

foreach  _row in D_ do

missing_count $\leftarrow$ find_missing(row);

total_count $\leftarrow$ length(row);

missing_rate $\leftarrow$ missing_count / total_count;

if  _missing_rate $\leq\theta$_ then

imputed_row$\leftarrow$ impute_model(row);

append(imputed_data, imputed_row);

else

append(imputed_data, row);

return _imputed_data_ ;

Algorithm 1 Imputation Algorithm for Missing Data

In this work, the linear regression imputation was selected due to its
superior performance in retaining the quality of the dataset, in contrast to
KNN Imputation and Iterative Imputation which resulted in a decline in the
$R^{2}$ score by $0.08-0.1$ in tests. Additionally, we chose to impute rows
which had 8 or less missing columns, which set a threshold for a row, where a
row must have at least 75% of its’ possibly missing data available in order to
have its’ remaining columns imputed. Dataset 2 had 36 columns and 33 of them
could have missing values. This allows for a good balance between the amount
of data imputed and the quality of data, since around 1,000 rows were saved
with this technique.
Reformatting Data for Analysis: Our primary goal here is to discern which
features most significantly influenced the shift in a country’s performance,
rather than focusing on specific yearly values. Given the distinct
characteristics of each dataset, tailored strategies were adopted to reformat
the original data into temporal multivariate timeseries datasets. For the
first and second datasets, apart from the country names, each row’s values
were deducted by the preceding row’s values, provided both rows pertained to
the same country. This process enabled the capture of year-to-year changes. On
the other hand, the third dataset, due to its unique structure, necessitated a
different approach. The only column where differences were computed was the
population. For other columns, values from the preceding row were directly
used to emphasize relative proportions rather than the absolute variations
across columns. It’s worth noting that these strategies have the potential to
be extended to various applications that possess a similar data format. Random
samples are shown in the Appendix to demonstrate the unique nature of
different datasets.

###  III-D Building Models with FLAML

Once this was done, the countries’ names were encoded into numerical values,
for the purpose of being able to use them with LIME later on. Then, the data
was split with a 80-20 split and trained through the use of the Python package
FLAML [5]. FLAML is built for machine learning, particularly for cost-
effective hyperparameter optimization for ensemble classification and
regression methods. The statistic that should be optimized, such as the
model’s RMSE or R2 value can be manually selected, as well as the amount of
time to be given to the algorithm and the algorithm will provide regular
updates on its’ progress. As seen in Fig 1., the models built by FLAML,
whether they’re with Light GBM or ExtraTrees provide a near-optimal fit for
the data, with the line of best fit being a nearly perfect diagonal line.
FLAML also leans towards specific models based on the data and will iterate
more on more suitable models, for example, for Dataset 2, FLAML tends to
always select Light GBM as the model used and for Dataset 3, FLAML tends to
avoid models like Random Forest. Within this work, we use FLAML in order to
train accurate and complicated models, in order to show the utility of LIME
and to ensure that LIME is truly being tested, not the abilities of FLAML.

(a) Predictions by Light GBM for Dataset 2

(b) Predictions by ExtraTrees for Dataset 3

Figure 1: Predictions by different models for Dataset 2 and Dataset 3

###  III-E Interpreting with LIME

While complex models, such as deep neural networks or ensemble methods
generated from FLAML, offer high accuracy, they often lack transparency,
making it challenging to understand their decision-making process. LIME
addresses this by locally approximating the black-box model’s predictions. For
a given prediction, LIME generates a set of perturbed data samples, obtains
predictions for these samples from the black-box model, and then fits a
simpler, interpretable model to these samples, typically a linear regression
model. LIME figures out what the most effective columns are through these
perturbations [15]. For example, if changing the value of the country’s area
greatly affects the prediction for the country’s population, then this might
be an important feature. Consequently, if changing a feature affects the
result by a minor amount or not at all, then the feature isn’t important.

For this purpose, we set $kernel\\_width$ to $1,0.849$, and $9000$ for three
datasets respectively, where the kernel width determines the width of the
neighborhood of points around the data point that’s allowed to influence the
local model and consequently, the explanation. The models built by LIME serve
as a proxy to the black-box model but is transparent in its decision-making.
In order to properly test the quality of LIME, we ran comprehensive tests,
where LIME made predictions for the entire test set in the case of the first
two datasets and for the first 250 rows in the case of the third test set,
while being restricted to a set number of columns, with that number being 3
for the first dataset and 10 for the other two datasets. The remainder of the
rows’ values were set to 0. The R2 values were collected and were compared to
cases where the same number of columns were randomly selected and the original
model’s R2 values. Therefore, we can show that LIME’s selection of columns is
significantly better than randomly selecting columns, as the brilliance of
LIME lies in its ability to provide insight into the black-box model’s
predictions for individual data points rather than a global approximation.

To further justify the results we receive from LIME, we compare its results
with those derived from Individual Conditional Expectation (ICE) plots [16],
which are a tool used for visualizing the model predictions for individual
instances as a function of specific input features. ICE plots are an extension
of Partial Dependence Plots (PDPs) [17], which show the average prediction of
a machine learning model as a function of one or two input features, holding
other features constant. As ICE plots offer a nuanced view of the relationship
between specific features and the predicted outcome, features with ICE lines
that show significant slopes suggest that a small change in the feature value
can lead to a substantial change in the prediction. A steep slope often
signifies feature importance, thus, as LIME assigns importance scores to
features, by comparing the top columns returned by LIME with the slopes and
variations observed in ICE plots and checking if frequently selected columns
have steep slopes, we can have improved confidence in whether LIME is
correctly identifying important features.

##  IV Results and Discussions

###  IV-A Experiment Setting

For each run of the FLAML algorithm, a time span of $T=180$ seconds is
allocated with the aim to optimize the $R^{2}$ value in our study. The
hyperparameters that are selected vary with each run, as well. The $R^{2}$
value, commonly known as the coefficient of determination, quantifies the
proportion of the variance in the dependent variable that can be predicted
from the independent variables. Essentially, it gauges the extent to which the
independent variables in a model elucidate the variability of the dependent
variable. The formula for $R^{2}$ is given by:

$R^{2}=\frac{\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}{\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}$
(4)

In the above equation:

  * •

$y_{i}$ denotes the observed value.

  * •

$\hat{y_{i}}$ symbolizes the value predicted by the model.

  * •

$\bar{y}$ represents the mean of the observed values.

The numerator encapsulates the squared discrepancies between the actual
observations and the predictions made by the model, reflecting the variance
that the model fails to explain. Meanwhile, the denominator computes the total
variance present in the data. The possible range for $R^{2}$ is between 0 and
1:

  * •

An $R^{2}$ value of 0 suggests that the model fails to explain any variability
of the dependent variable around its average.

  * •

An $R^{2}$ value of 1 indicates that the model accounts for all the
variability.

For Dataset 1, the models considered include Light GBM, Random Forest,
XGBoost, ExtraTrees, and a depth-limited XGBoost. In the case of Dataset 2,
only Light GBM is utilized, while for Dataset 3, all models except Random
Forest are employed. These are all variants that support regression. Model
selection is refined based on frequency: models that are either consistently
favored or rarely chosen are omitted, allowing the algorithm to focus on the
more influential models. Initially, long short-term memory networks (LSTMs)
are explored, but are later excluded due to their unsatisfactory $R^{2}$
preliminary results, even when using simpler LSTM configurations. A potential
rationale for this exclusion is that all three datasets have a relatively
small size, which might hinder the LSTM’s ability to construct a sturdy model.
The algorithm was run five times for each dataset, with the results being
provided in Table I. The mean $R^{2}$ values are included to give the viewer
an overall impression of the results.

Statsitic | Dataset 1 | Dataset 2 | Dataset 3
---|---|---|---
Run #1 | 0.092 | 0.924 | 0.988
Run #2 | 0.036 | 0.868 | 0.995
Run #3 | 0.21 | 0.846 | 0.994
Run #4 | 0.071 | 0.904 | 0.992
Run #5 | 0.226 | 0.883 | 0.996
Mean | 0.127 | 0.885 | 0.993
TABLE I: A table showing the $R^{2}$ values for each of the datasets with
FLAML.

###  IV-B LIME’s Results

To assess the effectiveness of LIME in feature selection, we carried out
multiple experiments, contrasting random column selection with LIME-guided
column selection. For each of the three datasets, these experiments were
conducted five times to mitigate the impact of randomness, with the average
results presented in Fig. 2. Remarkably, across all three sets of experiments,
LIME-guided feature selection consistently outperformed random selection in
terms of $R\textsuperscript{2}$ values. This was true even for Dataset 1,
which is the most difficult to understand. Moreover, there was an absolute
increase in the $R\textsuperscript{2}$ values in every instance. This was
particularly noteworthy for the third dataset, where the
$R\textsuperscript{2}$ values surged by an average of 0.4904 when employing
LIME for feature selection.

Figure 2: A bar graph comparing the $R^{2}$ values of selecting random columns
VS selecting columns with LIME.

This can be further verified with an one-sided Paired T-Test, which compares
the results without the use of LIME to the results with LIME. This test showed
significance for the first dataset at $\alpha$ = 0.02, while the latter two
results showed significance at $\alpha$ = 0.005, therefore, the approach that
was taken by LIME was clearly statistically significant and beneficial
compared to random predictions. Most notably, the P-value for Dataset 2 was
0.00005922. Dataset 1 was difficult to interpret and make predictions for, but
we’ve shown that some data and insights could be discovered, hence why the R2
were significant at $\alpha$ = 0.02, while Datasets 2 and 3 could be fully
interpreted by LIME and returned excellent results for the R2 values. LIME was
equally beneficial for individual predictions, as will be demonstrated through
the use of real-world scenarios and ICE Plots.

###  IV-C Case Studies

To illustrate LIME’s efficacy, we opted for a case study approach using two
extreme cases from Dataset 2. This approach not only validates the quality of
LIME but also lends real-world context to its utility. The key idea is to
match significant changes in a country’s economic freedom with major
historical or socio-political events and see if LIME’s explanations align with
these changes.
For instance, Syria experienced a substantial decline in its economic freedom
during 2011 and 2012, largely due to the onset of the Syrian civil war, a part
of the broader Arab Spring
movements111https://www.atlanticcouncil.org/blogs/menasource/the-economic-
collapse-of-syria/. In 2011, Syria’s actual economic freedom score plummeted
by 0.91 points. When subjected to LIME’s analysis, the algorithm estimated a
significant drop and associated this decline primarily with three factors:
inflation, credit market regulation, and regulatory trade barriers. Their
respective scores in the dataset decreased by 6.39, 1.4, and 0.56 points
between 2011 and 2012, which align with the observations found in Fig 3. It’s
worth noting that Syria faced an inflation rate of 36.7%
222https://www.macrotrends.net/countries/SYR/syrian-arab-republic/inflation-
rate-cpi in 2012 and also experienced severe disruptions in its business
environment, leading to a mass exodus of foreign investors and tourists. These
real-world factors align well with LIME’s explanation, further validating its
effectiveness.

Figure 3: A prediction made by LIME showing the columns that most contributed
to Syria’s drop in Economic Freedom between 2011 and 2012.

Inversely, Brazil had one of the largest improvements in the dataset, with a
strong comeback that took place from 1995 to 2000 and raised the country’s
economic freedom by 1.32 points. This was mainly in part due to previous
concerns that the country had in 1990 to 1995 being fixed. For example, Brazil
suffered from massive inflation, first in early 1990 and then in 1994, with
yearly rates of 2,947.73% and 2,075.89%, respectively, however, after the
Plano Real, a stabilization program, was introduced to curb this, the largest
yearly inflation from 1996-2000 was merely 15.76%, in 1996
333https://www.macrotrends.net/countries/BRA/brazil/inflation-rate-cpi. A
black market also existed, due to the ineffectiveness of actual money and the
need for affordable food and water. Finally, money growth also greatly
increased, which is calculated through the average annual growth of a
country’s money supply over the last five years minus average annual growth of
real GDP in the last ten years. Annual money growth was massive in 1994 and at
over 6,000% in 1994 444https://www.ceicdata.com/en/indicator/brazil/m2-growth,
but would never reach such heights again and remained below 30% during the
five-year period, largely in part because of the disastrous inflation being
fixed.

Figure 4: A prediction made by LIME showing the columns that most contributed
to the rise in Brazil’s Economic Freedom from 1995 to 2000.

###  IV-D Comparison with State-of-the-art Methods

In order to further justify the effectiveness of LIME, we used Individual
Conditional Explanation plots, acting as an extension of Partial Dependence
Plots, to allow for a further analysis of LIME by analyzing the slopes of each
feature. A feature with a steeper slope than other features in the ICE plot
will often indicate feature importance. Thus, by comparing the columns
provided by LIME with the ICE plots, we can assess the quality of LIME’s
predictions and show that the ICE plots verify our results.

In our study, ICE was employed to scrutinize changes at the level of
individual instances. To support this, as an example 20 rows were chosen using
LIME’s Submodular Pick function for Dataset 3, which assembles a series of
local, minimally redundant explanations. The selection of 20 rows aimed to
offer comprehensive coverage, align coherently with ICE plots that also focus
on individual rows, and yield conclusive insights. We generated Table II
detailing the frequency with which specific columns appeared among the top 5
chosen for these explanations. Limiting the selection to 5 columns, rather
than 10, helps to concentrate attention on the most influential features. Our
hypothesis posits that the features selected by LIME—due to its focus on
perturbing features to identify those with the most significant impact—will
corroborate their importance when examined through ICE plots, thus making them
ideal candidates for generating explanations.

Feature Name | Frequency
---|---
Area in Square Kilometers | 20
Sex Ratio at Birth | 17
Infant Mortality Rate for Females | 14
Under Age 5 Mortality for Females | 12
Age 1-4 Mortality for Females | 9
Age-Specific Fertility Rate 20-24 | 7
Infant Mortality Rate for Both Sexes | 4
Age 1-4 Mortality for Both Sexes | 3
Crude Death Rate | 3
Infant Mortality Rate for Males | 2
Age-Specific Fertility Rate 25-29 | 2
Age-Specific Fertility Rate 35-39 | 2
Under Age 5 Mortality for Males | 1
Under Age 5 Mortality for Both Sexes | 1
Age-Specific Fertility Rate 40-44 | 1
Life Expectancy at Birth for Males | 1
Life Expectancy at Birth for Females | 1
TABLE II: A frequency table of how LIME selects features in Dataset 3.

From Table II, it is obvious that even with the algorithm trying to make the
explanations as varied as possible, the feature Area in Square Kilometers was
selected every single time, while Sex Ratio at Birth, Infant Mortality Rate
for Females and Under Age 5 Mortality for Females were also commonly selected.

Figure 5: ICE Plots showing how greatly the columns that are most selected by
LIME affect the change in population, for Dataset 3.

It’s clear that these selected columns are backed up by the ICE Plots in Fig.
5, which all have large variances and can affect the change in a country’s
population by several hundred thousand. We also noted that the features Area
in Square Kilometers and Sex Ratio at Birth have the steepest slopes. In
conclusion, LIME and ICEs both agree that these columns were the most
important to Dataset 3’s predictions.

##  V Conclusion

In summary, this research presents a rigorous exploration into the use of LIME
for multivariate temporal analysis on global annual datasets. Our approach
successfully identifies key factors contributing to temporal shifts, offering
explanations that are critical for both academic inquiry and practical
applications. Through the integration of LIME with other advanced methods such
as ICE plots, we have not only validated the efficacy of our approach but also
opened new avenues for explainable AI in temporal analysis.

The implementation of multiple imputation techniques to handle missing data in
the second dataset exemplifies the adaptability and robustness of our method.
Approximately 65% of the records in this dataset had missing values, and our
strategy ensured reliable, high-quality imputations that enriched the overall
analysis.

Furthermore, our study lays the groundwork for future research in this area.
The analytical frameworks presented here can be extended to tackle more
complex datasets and questions, offering transformative impacts that range
from the reshaping of governmental policies to steering economic strategies.

Future research could focus on comparing the effectiveness of LIME against
other explainability models or delve into real-world applications that could
benefit from our approach.

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##  Appendix A Dataset Snippets

The following tables display some randomly selected records from each dataset,
to provide an understanding of the challenges faced with the missing data in
Dataset 2 and an idea of what kind of data was analyzed within the paper.

Country Name | Year | Life Ladder | Log GDP per Capita | Social Support | Life Expectancy | … | Dystopia + Residual
---|---|---|---|---|---|---|---
Gabon | 2021 | 5.075 | 9.533 | 0.754 | 58.25 | … | 2.201
Honduras | 2016 | 5.648 | 8.573 | 0.774 | 61.725 | … | 2.296
Poland | 2020 | 6.139 | 10.39 | 0.953 | 68.875 | … | 2.056798
Togo | 2022 | 4.239 | 7.685 | 0.579 | 57.7 | … | 2.061
United States | 2016 | 6.804 | 10.985 | 0.897 | 66.475 | … | 2.728
Zambia | 2018 | 4.041 | 8.139 | 0.718 | 53.975 | … | 1.667
TABLE III: A snippet of Dataset 1. Year | Country | Economic Freedom | … | 5a_credit_market_reg | 5b_labor_market_reg | 5c_business_reg
---|---|---|---|---|---|---
2012 | Australia | 7.96 | … | 9.596215 | 6.744523 | 8.036912
1995 | Bangladesh | 5.3 | … | 5.122456 | N/A | N/A
2003 | Benin | 6.03 | … | 9.167872 | 4.665644 | N/A
1990 | Switzerland | 8.21 | … | 8.153153 | 5.4835 | N/A
2012 | Tanzania | 6.62 | … | 8.985725 | 6.477967 | 5.588262
1995 | Uganda | 5.09 | … | 4.285619 | N/A | N/A
2011 | Syria | 6.18 | … | 6.981081 | 4.841801 | 5.970616
2012 | Syria | 5.27 | … | 5.585419 | 5.432596 | 4.987837
1995 | Brazil | 4.66 | … | 4.573693 | 5.296847 | 4.942583
2000 | Brazil | 5.98 | … | 6.228699 | 4.82906 | 5.470623
TABLE IV: A snippet of Dataset 2. Country/Area Name | Year | Area in KM2 | Population | … | Crude Death Rate | Net Migration Rate
---|---|---|---|---|---|---
Andorra | 2010 | 468 | 84,563 | … | 5.89 | 0
Cambodia | 2022 | 176,515 | 16,713,015 | … | 5.76 | -2.7
Indonesia | 2023 | 1,811,569 | 279,476,346 | … | 6.77 | -0.71
Mauritania | 2012 | 1,030,700 | 3,356,215 | … | 8.77 | -0.89
Spain | 2005 | 498,980 | 43,704,367 | … | 8.86 | 14.55
United States | 2023 | 9,150,541 | 339,665,118 | … | 8.4231 | 3.0116
TABLE V: A snippet of Dataset 3.