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# Evaluating Machine Learning Performance Using Python for Neural Network Models in Urban Transportation in New York City Case Study

Mohsen Mohammadagha, Saeed Asadi, Hajar Kazemi Naeini

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Mohsen Mohammadagha, Saeed Asadi, Hajar Kazemi Naeini. Evaluating Machine Learning Performance Using Python for Neural Network Models in Urban Transportation in New York City Case Study. 2025. hal-05031543

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# Evaluating Machine Learning Performance Using Python for Neural Network Models in Urban Transportation in New York City Case Study

Mohsen Mohammadagha¹*, Saeed Asadi², Hajar Kazemi Naeini².

¹, Ph.D. Student at Department of Civil Engineering, University of Texas at Arlington, Arlington, Texas, USA.

², Master's Student, Civil Engineering Department, University of Texas at Arlington, Arlington, Texas, USA.

* Correspondence: mxm4340@mavs.uta.edu (M.M);

## Abstract:

This study evaluates machine learning methods, focusing on neural networks, to address the XOR classification problem and taxi trip duration prediction in New York City. Addressing the need for robust predictive models, the research developed a custom neural network library with features like forward and backward propagation, Sigmoid and Tanh activation functions, and binary cross-entropy and mean squared error loss functions. The XOR model achieved a significant reduction in loss from 0.7389 to 0.0154 over 10,000 epochs, with predictions closely approximating expected outputs. For the taxi trip duration task, three architectures—small (RMSLE: 0.474), medium (RMSLE: 0.455), and large (RMSLE: 0.462)—were evaluated, with the medium model achieving the best balance of performance and efficiency. These results highlight the importance of feature engineering and architecture selection in achieving accurate predictions. Future research should explore advanced techniques like Neural Architecture Search (NAS) and hybrid models to enhance scalability across fields such as transportation, healthcare, environmental impact, and energy optimization.

**Keywords:** Machine Learning, Neural Networks, Python Programming, New York City Case Study, Taxi Trip Duration, Urban Transportation.

## 1 Introduction

Neural networks have demonstrated remarkable potential in optimizing [1] critical processes and solving complex problems across various domains, like healthcare, sports [2], extending the lithium-ion batteries' lifespan [3], space exploration, and renewable energy. In healthcare, neural networks are transforming diagnostics and treatment planning by analyzing medical images, predicting patient outcomes, and personalizing care. For instance, vision transformers in convolutional neural networks (CNNs) [4] are widely used to interpret X-rays, CT scans [5], physiological signals analysis [6] and MRIs with high accuracy, aiding in the early detection of diseases like cancer or cardiovascular conditions, and supporting structural health monitoring [7] for infrastructure. Beyond diagnostics, neural networks also enhance hospital [8] management by predicting patient admissions [9], optimizing resource allocation for supply chain analytics [10], and improving

workflow efficiency with machine learning [11]. These applications underscore neural networks' versatility in addressing clinical and operational challenges in healthcare. In the realm of research analytics, neural networks are redefining how researchers strategize and evaluate performance, assess how performance loss impacts key performance indicators (KPIs) [12], and apply hybrid Neural Networks for optimizing smart buildings [13], [14] by using estimation in energy systems, and bifacial modules [15] in PV power.

By integrating machine learning and probabilistic modeling with software like ABAQUS [16] for assessing pipelines corrosion [17], ETAS Model for probabilistic seismic hazard assessment [18] MICP method for soil bio-improvement [19], road fixture monitoring [20] large-scale mixed-integer convex program (MICP) for real-time optimization [21] or an Artificial Neural Network (ANN), it is possible to analyze and predict behaviors under specific conditions,
particularly in the civil engineering industry, such as predicting damage for Trusses [22], the effect of steel plate [23], [24] in structural damage[25], pipeline conditions analysis[26] [27]or prediction [28], [29] of remaining useful life of pipes [30].

Predictive models analyze historical data to forecast match outcomes [31] or assess player performance, providing actionable insights for training and game preparation. Advanced techniques such as deep learning and numerical approaches like those used in earth-to-air heat exchangers evaluations [32] combined with optimization algorithms, are being employed across various fields, including finance, healthcare and sports. These methods are applied to classify sports images [33] or analyze player movements [34] in real-time, offering coaches valuable feedback on tactics and individual skills. Similarly, ML-accelerated method [35], gradient-based optimization techniques [36], hybrid physics-based and data-driven approaches in real time, enhance predictive analysis across energy systems [37], address sustainability challenges in urban planning [38] and improve structural evaluations. Additionally artificial neural networks (ANNs) are utilized to forecast satellite surface temperature rates [39], while hybrid simulation [40] integrate experimental and numerical models to advance structural analysis.

By processing vast datasets from telescopes or satellites, these algorithms help identify celestial patterns or anomalies that might otherwise go unnoticed. These methods reflect the growing influence of neural networks in unlocking new possibilities across diverse fields. In addition, other methods of machine learning like the ultra-high dimensionality and strong nonlinearity inherent in CNNs [41] render them resistant to traditional gradient-based optimization techniques, like GNN and Deep GCN model's robustness and scalability [42], across various parameters[43].

The development of neural network libraries has become a cornerstone in advancing machine learning applications, enabling researchers to design and experiment [44]with custom architectures tailored to specific

problems. Among these challenges, the XOR problem has long served as a benchmark for testing the capabilities of neural networks. This problem, characterized by its non-linear separability, requires sophisticated architectures and activation functions to achieve accurate classification. The XOR problem's [45] simplicity yet computational complexity makes it an ideal test case for evaluating the performance of neural network frameworks and their underlying components.

The implementation of a custom neural network library involves creating foundational components such as layers, activation functions, and loss functions, each contributing to the network's ability to learn and generalize. Central to this process is the design of a Layer class that supports forward and backward propagation, allowing for efficient computation across various architectures. Activation functions like Sigmoid[46], ReLU, and Tanh play a crucial role in introducing non-linearity into the model, while loss functions such as binary cross-entropy measure the performance of predictions against actual outputs. Together, these elements form the building blocks of a robust neural network capable of solving complex problems like XOR.

Beyond solving XOR, neural networks have been increasingly applied to real-world problems such as predicting taxi trip durations. This task involves preprocessing large datasets to extract meaningful features, normalizing data for consistency, and experimenting with different architectures to optimize performance. Features such as pickup and drop-off locations, trip distance[47], and temporal variables are transformed and analyzed to improve predictive accuracy. The iterative process of model selection, hyperparameter tuning, and validation demonstrates the adaptability of neural networks in addressing diverse challenges.

The comparison of activation functions highlights their impact on training efficiency and model performance. For instance, while Sigmoid functions are effective in certain scenarios, Tanh [48] often outperforms it in tasks requiring faster convergence due to its wider output range. Similarly, ReLU has gained
popularity for its simplicity and computational efficiency in deeper networks. These insights underline the importance of selecting appropriate activation functions based on the problem's characteristics and computational constraints.

This study aims to explore the implementation of a custom neural network library and its application to both theoretical (XOR) and practical (taxi trip duration) problems. By examining different architectures, activation functions, and preprocessing techniques, this research seeks to contribute valuable insights into the design and optimization of neural networks. The findings not only validate the efficacy [49] of custom-built frameworks but also provide a sample for future advancements in machine learning applications.

## 2 Methodology

The methodology for this study involves the implementation of a custom neural network library [50] and the application of this library to solve both theoretical and practical machine learning problems, specifically the XOR classification problem and taxi trip duration prediction. The process is divided into multiple stages, including library development, dataset preprocessing, model training, evaluation, and hyperparameter optimization [51].

The first step was the development of a neural network library from scratch. A foundational Layer class was created to define essential forward and backward propagation methods. This class served as a blueprint for implementing specific layers such as Linear (for weight multiplication and bias addition) and activation functions like Sigmoid, Tanh, and ReLU. A Sequential class was also implemented to manage multiple layers, enabling efficient forward and backward propagation through the entire network. The library included functionality for saving and loading model weights, facilitating model reuse without retraining.

The mathematical foundation of the neural network library begins with the implementation of linear layers, which form the basis for weight[52] multiplication and bias addition. The forward pass in a linear layer is computed using the formula in equation 1:

$$Output = Input \cdot W^T + b \quad (1)$$

Where W represents the weight matrix, b is the bias vector, and T denotes the transpose operation. During backpropagation, gradients of the weights (dW) and biases (db) are calculated to update parameters using gradient descent. The formulas for these gradients are as shown in equation 2:

$$\begin{aligned} dW &= GradOutput^T \cdot Input, \\ db &= \sum(\{GradOutput\}) \end{aligned} \quad (2)$$

These gradients are then used to update parameters as follows:

$$W = W - \eta \cdot dW, \quad b = b - \eta \cdot db \quad (3)$$

Where η is the learning rate. Activation functions introduce non-linearity into the network, enabling it to learn complex patterns. For example, the Sigmoid activation function [53] is defined in equation 4:

$$\sigma(x) = \frac{1}{1 + e^{-x}} \quad (4)$$

It's derivative, used during backpropagation, shown in equation 5:

$$\sigma'(x) = \sigma(x) \cdot (1 - \sigma(x)) \quad (5)$$

Similarly, the Tanh activation function is expressed as shown in equation 6:

$$\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \quad (6)$$

With its derivative given by equation 7:

$$\tanh'(x) = 1 - (\tanh(x))^2 \quad (7)$$

These derivatives are essential for propagating errors through non-linear layers during training, for tasks like XOR classification and taxi trip duration prediction, loss functions measure model performance. Binary cross-
entropy [54] loss for binary classification problems is calculated as equation 8:

$$L = -\frac{1}{N} \sum_{i=1}^{N} [y_i \log(\hat{y}_i) + (1 - y_i) \log(1 - \hat{y}_i)] \quad (8)$$

where $y_i$ is the true label and $\hat{y}_i$ is the predicted probability. For regression tasks like trip duration prediction, mean squared error (MSE) loss [55] is used in equation 9:

$$L = \frac{1}{N} \sum_{i=1}^{N} (\hat{y}_i - y_i)^2 \quad (9)$$

The gradients of these loss functions [56] with respect to predictions are computed during backpropagation to adjust model parameters effectively.

The XOR problem was chosen as a benchmark to test the library's capabilities. A small neural network with one hidden layer was constructed using the library. The network architecture consisted of two input neurons, two hidden neurons with a chosen activation function (e.g., Sigmoid or Tanh), and one output neuron with a Sigmoid activation function. Binary cross-entropy loss was used as the objective function, and gradient descent was employed for optimization. The model was trained over 10,000 epochs with periodic monitoring of the loss to ensure convergence.

<table><caption>Table 1, Benchmark Comparison,</caption><thead><tr><th>Model</th><th>Network Size</th><th>Learning Rate</th><th>Epochs</th><th>Early Stop E.</th><th>Test RMSLE</th></tr></thead><tbody><tr><td>Model 1</td><td>Small</td><td>0.001</td><td>50</td><td>19</td><td>0.474</td></tr><tr><td>Model 2</td><td>Medium</td><td>0.0005</td><td>100</td><td>43</td><td>0.455</td></tr><tr><td>Model 3</td><td>Large</td><td>0.0005</td><td>200</td><td>37</td><td>0.462</td></tr></tbody></table>

The benchmark model used a 3-layer neural as shown in Table 1, network with ReLU activation and trained for 7 epochs, achieving an RMSLE of 0.513. For taxi trip duration prediction, a larger dataset was utilized, requiring extensive preprocessing. Key features

such as pickup and drop-off coordinates, passenger count, trip distance (calculated using the Haversine formula), and time-based variables (e.g., pickup hour, day, month) were extracted. Numerical features were normalized using standard scaling to ensure consistency across varying data ranges. The dataset was split into training, validation, and test sets to evaluate model performance effectively.

Three neural network architectures—small, medium, and large—were designed for the taxi trip duration task. These models varied in the number of layers and neurons per layer to assess the impact of depth and complexity on performance. Mean squared error (MSE) loss was used as the objective function for regression tasks. Early stopping based on validation loss was implemented to prevent overfitting during training.

The dataset [57] in Table 2, contains several key features describing taxi trips. No. 1, pickup_longitude, and No. 2, pickuplatitude, represent the geographic coordinates (longitude and latitude) where the taxi trip started, while No. 3, dropoff_longitude, and No. 4, dropoff latitud, indicate the coordinates where the trip ended. No. 5, the passenger_count column, shows the number of passengers on each trip, which can range from one to the maximum capacity of a taxi. No. 6, the trip_duration, represents the total time of the ride, typically measured in seconds or minutes, and No. 7, the trip_distance, indicates how far the taxi traveled during the trip, usually measured in miles or kilometers. Together, these features provide essential information about trip origins, destinations, passenger details, and travel metrics.

<table><caption>Table 2, Summary Statistics for 1,312,779 samples data</caption><thead><tr><th>No</th><th>Mean</th><th>Std</th><th>Min</th><th>Median</th><th>Max</th></tr></thead><tbody><tr><td>1</td><td>-73.975</td><td>7.35E-02</td><td>-121.933</td><td>-73.982</td><td>-61.336</td></tr><tr><td>2</td><td>40.751</td><td>3.29E-02</td><td>34.359</td><td>40.754</td><td>51.881</td></tr><tr><td>3</td><td>-73.973</td><td>7.32E-02</td><td>-121.933</td><td>-73.98</td><td>-61.336</td></tr><tr><td>4</td><td>40.752</td><td>3.58E-02</td><td>32.181</td><td>40.755</td><td>43.921</td></tr><tr><td>5</td><td>1.66</td><td>1.31</td><td>0</td><td>2</td><td>9</td></tr><tr><td>6</td><td>958</td><td>5,409</td><td>1</td><td>662</td><td>3,526,282</td></tr></tbody></table>
<table><tr><td>7</td><td>3.44</td><td>4.3</td><td>0</td><td>2.09</td><td>1,240</td></tr></table>

Visualization techniques were employed to analyze feature distributions and relationships within the dataset. Scatter plots highlighted geographic patterns in pickup and drop-off locations, while heatmaps revealed correlations between numerical features. These insights guided feature selection and informed decisions about model architecture.

In machine learning, an epoch as shown in Table 3, is one complete pass through the entire training dataset during model training. Loss measures the model's error or how far its predictions are from the actual values. Lower loss indicates better performance, as seen in the table where loss decreases with more epochs.

<table><caption>Table 3, Training Progress: Epoch vs. Loss, (Training XOR with sigmoid activation in the hidden layer).</caption><thead><tr><td>Epoch</td><td>Loss</td></tr></thead><tbody><tr><td>0</td><td>0.7389</td></tr><tr><td>1000</td><td>0.6793</td></tr><tr><td>2000</td><td>0.4663</td></tr><tr><td>3000</td><td>0.1384</td></tr><tr><td>4000</td><td>0.0645</td></tr><tr><td>5000</td><td>0.0403</td></tr><tr><td>6000</td><td>0.029</td></tr><tr><td>7000</td><td>0.0225</td></tr><tr><td>8000</td><td>0.0183</td></tr><tr><td>9000</td><td>0.0154</td></tr></tbody></table>

Table 4, illustrates the training of a neural network to solve the XOR problem, as part of a neural network assignment. It employs a sigmoid activation function in the hidden layer. Over 10,000 epochs, the loss declines remarkably from 0.7389 to 0.0154, signaling successful learning. Final predictions closely approximate expected XOR outputs: near 0 for (0,0) and (1,1), and near 1 for (0,1) and (1,0), demonstrating reliability and precision.

<table><caption>Table 4, Predictions on XOR input:</caption><thead><tr><td>Input Pair</td><td>Prediction</td></tr></thead><tbody><tr><td>(0, 0)</td><td>0.01293745</td></tr><tr><td>(0, 1)</td><td>0.98262338</td></tr><tr><td>(1, 0)</td><td>0.98831346</td></tr><tr><td>(1, 1)</td><td>0.01091259</td></tr></tbody></table>

The models were evaluated based on metrics such as root mean squared logarithmic error (RMSLE) for regression tasks and binary accuracy for classification tasks. For XOR classification, final predictions were compared against expected outputs to validate the model's reliability. For taxi trip duration prediction, training and validation loss curves were analyzed to assess convergence and detect overfitting.

Finally, benchmark comparisons were conducted to evaluate the custom models against pre-existing baselines. The results demonstrated that medium-sized architectures provided an optimal balance between performance and computational efficiency for the taxi dataset. The study concluded with a discussion of findings and recommendations for future improvements in neural network design and application.

## 3 Results

The results of this study highlight the successful implementation and application of a custom neural network library to address both the XOR classification problem and a real-world regression task involving taxi trip duration prediction. For the XOR problem, the neural network demonstrated effective learning, achieving a significant reduction in binary cross-entropy loss over 10,000 epochs. The final predictions closely matched the expected XOR outputs, validating the library's capability to model non-linear relationships using activation functions like Sigmoid and Tanh. In the taxi trip duration task, extensive preprocessing, including feature engineering
and normalization, was performed on a large dataset to extract meaningful variables such as trip distance and temporal features. Three neural network architectures (small, medium, and large) were trained using mean squared error as the loss function, with early stopping implemented to prevent overfitting. The medium-sized model achieved the best performance, outperforming the benchmark model in terms of root mean squared logarithmic error (RMSLE), demonstrating the efficacy of the custom library in solving complex regression problems.

<figure><img src="image_3.png" /><figcaption>Figure 1, Small Model Loss</figcaption></figure>

The Small Model Loss shows in Figure 1, consisting of a single dense layer with 50 neurons, demonstrates rapid convergence within the first 10 epochs. This is evident from the steep decline in both training and validation loss during the initial stages of training. However, the model's simplicity limits its capacity to generalize complex patterns, as indicated by the relatively higher Mean Squared Error (MSE) loss compared to larger models. Early stopping was triggered at epoch 19, suggesting that the model reached its optimal performance quickly but lacked further improvement. The test Root Mean Squared Logarithmic Error (RMSLE) of 0.474 highlights the trade-off between computational

efficiency and predictive accuracy for this smaller architecture.

<figure><img src="image_4.png" /><figcaption>Figure 2, Medium Model Loss</figcaption></figure>

The Medium Model Loss graph shows in Figure 2, incorporates two dense layers (100 and 50 neurons), enabling it to capture more complex relationships within the data. This is reflected in its lower validation loss and smoother convergence compared to the small model. Training continued for 43 epochs before early stopping was triggered, indicating that the additional layer and increased neuron count allowed for more gradual optimization. The medium model achieved a test RMSLE of 0.455, outperforming both the small model and the benchmark. This balance between complexity and performance makes it an ideal choice for this task, as it effectively minimizes overfitting while leveraging its enhanced capacity.

The Large Model Loss graph shows in Figure 3, featuring three dense layers (200, 100, and 50 neurons), exhibits a similar convergence pattern but with slightly higher validation loss compared to the medium model. Early stopping occurred at epoch 37, which is earlier than the medium model's stopping point, suggesting
diminishing returns from adding more layers
and neurons.

<figure><img src="image_5.png" /><figcaption>Figure 3, Large Model Loss</figcaption></figure>

While this architecture achieved a test RMSLE of 0.462—better than the small model but slightly worse than the medium model—it required more computational resources without significant performance gains. This indicates that while deeper networks can capture additional complexity, they may also introduce redundancy or overfitting if not carefully managed.

The bar chart in Figure 4, illustrates the distribution of passenger counts in a dataset. Most trips involve a single passenger, as indicated by the highest frequency for a count of 1.

<figure><img src="image_6.png" /><figcaption>Figure 4, Distribution of Passenger Count</figcaption></figure>

The frequency decreases significantly for higher passenger counts, with very few trips having more than 2 passengers. This highlights a strong skew toward solo travelers.

The histogram Figure 5, displays the distribution of the logarithm of trip durations. The data forms a bell-shaped curve, indicating a near-normal distribution. Most log-transformed trip durations fall between 6 and 8, with a peak around 7. Outliers are present on both ends, but their frequency is minimal compared to the central values.

<figure><img src="image_7.png" /><figcaption>Figure 5, Distribution of Log Trip Duration</figcaption></figure>

The scatter plot in Figure 6, visualizes pickup locations based on latitude and longitude. A dense cluster is observed around a specific region, likely representing a major urban area with high activity. Sparse points outside this cluster indicate occasional pickups in less frequented areas. The plot highlights the geographic concentration of pickups, emphasizing urban dominance.

<figure><img src="image_8.png" /><figcaption>Figure 6, Pickup Locations</figcaption></figure>
This scatter plot in Figure 7, shows drop-off locations based on latitude and longitude. A dense cluster is visible around a specific region, likely representing a high-activity urban area. Sparse points outside the cluster indicate occasional drop-offs in less frequented areas. This visualization highlights the geographic concentration of drop-offs, similar to pick up locations.

<figure><img src="image_9.png" /><figcaption>Figure 7, Dropoff Locations</figcaption></figure>

This bar chart in Figure 8, depicts the hourly distribution of pickups throughout the day. Pickup activity is lowest during early morning hours (3–5 AM) and increases steadily, peaking around 6–8 PM. A secondary smaller peak occurs at midnight. This pattern suggests higher demand during commuting hours and evenings, with reduced activity during late-night and early-morning hours.

<figure><img src="image_10.png" /><figcaption>Figure 8, Hourly Distribution of Pickups</figcaption></figure>

This box plot in Figure 9 shows the relationship between passenger count and the log-transformed trip duration. Most trips involve 1-2 passengers, with similar median trip durations

across these counts. Outliers are present for all passenger counts, indicating occasional long trips. Trips with rare passenger counts (like 6 or more) show more variability and fewer data points, suggesting less frequent occurrences.

<figure><img src="image_11.png" /><figcaption>Figure 9, Passenger Count vs Log Trip Duration</figcaption></figure>

This correlation heatmap in Figure 10, illustrates relationships among variables such as pickup/dropoff coordinates, passenger count, and trip duration. Pickup and dropoff longitudes exhibit a weak negative correlation (-0.03), while latitudes show minimal correlation with other variables. Passenger count and trip duration are nearly uncorrelated with spatial features, indicating independence. Overall, the heatmap highlights weak interdependence across most variables, suggesting that they contribute distinct information to the dataset.

<figure><img src="image_12.png" /><figcaption>Figure 10, Correlation Heatmap</figcaption></figure>

Figure 11, shows the distribution of passenger counts across trips, showing that the majority of
trips involve a single passenger, while significantly fewer trips accommodate larger groups. This indicates a strong skew toward solo travelers, which may reflect individual preferences or the convenience of ride-hailing services for single riders. The second image is a correlation heatmap that visualizes the relationships between various features in the dataset. The analysis reveals generally weak correlations among variables. For example, pickup and dropoff coordinates show minimal correlation (-0.03), while passenger count and trip duration exhibit near-zero correlations with other features, highlighting their independence. This suggests each feature contributes unique, uncorrelated information to the dataset.

<figure><img src="image_13.png" /><figcaption>Figure 11, Distribution of Passenger Count</figcaption></figure>

The violin plot in Figure 12, visualizes the distribution of log-transformed trip durations across pickup hours. The distributions are consistent throughout the day, with most durations centered around similar medians, though variability and outliers are present.

The correlation heatmap reveals weak relationships among variables, with pickup and dropoff coordinates showing minimal correlation, and passenger count and trip duration displaying near-zero correlations with other features, indicating their independence.

<figure><img src="image_14.png" /><figcaption>Figure 12, Violin Plot of Log Trip Duration by Pickup Hour</figcaption></figure>

The scatter plot in Figure 13, shows the relationship between passenger count and log-transformed trip duration. There is no clear trend, as the durations remain consistent across passenger counts, suggesting minimal influence of passenger count on trip duration. The correlation heatmap further confirms weak relationships among variables, with near-zero correlations for passenger count and trip duration with other features, highlighting their independence. Pickup and dropoff coordinates also exhibit minimal correlations.

<figure><img src="image_15.png" /><figcaption>Figure 13, Regression: Log Trip Duration vs Passenger Count</figcaption></figure>

This line plot in Figure 14, depicts the monthly average log-transformed trip durations over six months. The trend shows a steady increase, starting at 6.84 in January and peaking at 6.92 in May and June.
<figure><img src="image_16.png" /><figcaption>Figure 14, Monthly Average Log Trip Duration</figcaption></figure>

This suggests that average trip durations gradually rise over the months, stabilizing by mid-year. The second plot includes a grid for enhanced readability, aiding in visualizing the incremental changes more clearly.

The bar chart in Figure 15, shows the count of trips by vendor ID, with Vendor 2 slightly outpacing Vendor 1 in trip volume, indicating a balanced but slightly higher activity for Vendor 2. The second line plot depicts the monthly average log-transformed trip duration, showing a steady increase from January (6.84) to June (6.92), suggesting longer trips as the year progresses.

<figure><img src="image_17.png" /><figcaption>Figure 15, Count of Trips by Vendor ID</figcaption></figure>

The plot in Figure 16, shows the distribution of trip distances, with most trips concentrated under 50 km, while longer distances are rare, creating a highly skewed distribution. The second plot illustrates the monthly average log-transformed trip duration, which steadily increases from January (6.84) to June (6.92),

indicating a gradual rise in trip durations over time.

<figure><img src="image_18.png" /><figcaption>Figure 16, Distribution of Trip Distance (km)</figcaption></figure>

The box plot Figure 17, compares passenger counts by vendor ID. Vendor 2 shows a broader range and higher median passenger count, while Vendor 1 has minimal variation. Outliers are present for both vendors, with some trips involving up to 8 passengers. The second line plot displays the monthly average log-transformed trip duration, showing a steady increase from January (6.84) to June (6.92), indicating longer trips over time.

<figure><img src="image_19.png" /><figcaption>Figure 17, Box Plot of Passenger Count by Vendor ID</figcaption></figure>

The heatmap in Figure 18, visualizes average trip durations by pickup hour and day of the week. Longer durations are observed during late-night and early-morning hours, particularly on weekends (like, 1,459.7 at 23:00 on Saturday). The line plot shows a steady increase in monthly average log-transformed trip durations, rising from 6.84 in January to 6.92
by May and June, indicating a gradual
lengthening of trips over time.

<figure><img src="image_20.png" /><figcaption>Figure 18, Average Trip Duration by Hour and Day of Week</figcaption></figure>

The histogram in Figure 19, visualizes the distribution of the log-transformed trip durations log (Trip Duration+1)) for the training dataset. The distribution is approximately normal, with a peak around 6. This transformation reduces skewness in the original trip duration data, making it more suitable for modeling and analysis. A few outliers are visible on the right, representing trips with unusually long durations.

<figure><img src="image_21.png" /><figcaption>Figure 19, Distribution of Log (Trip Duration + 1) (Train)</figcaption></figure>

The scatter plot in Figure 20, displays sample pickup locations based on longitude and latitude, providing insights into the spatial distribution of taxi activity. The dense cluster visible in the plot corresponds to Manhattan, New York City, a central hub for taxi pickups

due to its high population density and vibrant
economic activity.

<figure><img src="image_22.png" /><figcaption>Figure 20, Sample Pickup Locations</figcaption></figure>

Sparse points outside this cluster represent pickups occurring in less frequented areas, such as the outer boroughs or suburban regions. This geographic distribution emphasizes the concentration of taxi pickups in urban areas, particularly Manhattan, where demand is highest. It highlights the city's role as a focal point for transportation services.

<figure><img src="image_23.png" /><figcaption>Figure 21, Feature Correlation Heatmap</figcaption></figure>

The heatmap in Figure 21, illustrates correlations among key features in the dataset. Strong positive correlations are observed between pickup and dropoff longitudes (0.80) and latitudes (0.49), reflecting geographic proximity. Trip distance shows a moderate
positive correlation with pickup longitude (0.25) and weak negative correlation with pickup latitude (-0.20). Passenger count, pickup hour, day, and month exhibit near-zero correlations with other variables, indicating their independence. Overall, spatial features are more interrelated than temporal or trip-specific features.

The first heatmap in Figure 10, focuses on spatial and trip-specific features, such as pickup and dropoff coordinates, passenger count, and trip duration, offering a concise view of their relationships. In contrast, the second heatmap in Figure 21, incorporates additional temporal dimensions, including pickup hour, day, and month, as well as trip distance. This broader perspective in the second heatmap allows for analyzing both temporal and spatial interactions. Correlation values differ slightly between the two due to the inclusion of different variables.

<figure><img src="image_24.png" /><figcaption>Figure 22, Passenger Count vs. Log (Trip Duration + 1)</figcaption></figure>

The scatter plot in Figure 22, visualizes the relationship between passenger count and the log-transformed trip duration. The data points are scattered without a clear trend, indicating that passenger count has little to no impact on trip duration. Most trips involve 1–2 passengers, with durations spanning a wide range. Outliers are present for higher passenger counts, but these are relatively rare.

The findings of this study underscore the effectiveness of the custom neural network library in addressing diverse machine learning challenges. For the XOR problem, the model achieved precise predictions, with outputs closely approximating expected values, demonstrating its ability to handle non-linear relationships. In the taxi trip duration task, the medium-sized model emerged as the optimal architecture, balancing performance and computational efficiency. The results also highlighted the importance of feature engineering and preprocessing in achieving robust predictions. Overall, the custom library outperformed benchmark models, showcasing its adaptability and reliability across classification and regression tasks.

# 4 Discussion

The results of this study provide a comprehensive evaluation of the capabilities of a custom neural network library in solving both theoretical and practical machine learning problems. The XOR problem served as a critical benchmark for assessing the library's ability to model non-linear relationships. By employing activation functions such as Sigmoid and Tanh, the network effectively captured the complexity of XOR's non-linear separability, achieving high accuracy over 10,000 epochs. This success highlights the importance of activation functions in introducing non-linearity to neural networks, enabling them to solve problems that linear models cannot address. Furthermore, the iterative process of forward and backward propagation demonstrated how gradient descent optimizes weights and biases to minimize loss functions like binary cross-entropy. These foundational principles not only validated the library's design but also showcased its potential for broader applications.

In contrast to XOR's theoretical simplicity, predicting taxi trip durations presented a more complex real-world challenge requiring advanced data preprocessing and feature
engineering. The dataset included diverse features such as geographic coordinates, trip distance, passenger count, and temporal variables like pickup hour and day. Feature scaling ensured consistency across numerical variables, while visualization techniques provided valuable insights into data distribution and correlations. For instance, scatter plots revealed geographic clustering in urban areas like Manhattan, while heatmaps highlighted weak interdependencies among most features. These analyses informed decisions about feature selection and model architecture, emphasizing the role of exploratory data analysis in machine learning workflows.

The comparison of three neural network architectures—small, medium, and large—offered valuable insights into the trade-offs between complexity and performance. While the small model demonstrated rapid convergence due to its simplicity, it lacked the capacity to generalize complex patterns effectively. On the other hand, the large model introduced redundancy and required greater computational resources without significant performance gains. The medium-sized model struck an ideal balance by leveraging sufficient complexity to capture relationships while avoiding overfitting. This architecture achieved the lowest RMSLE score among all models tested, outperforming both smaller networks and pre-existing benchmarks.

These findings also underscore the importance of early stopping as a regularization technique to prevent overfitting during training. By monitoring validation loss, early stopping ensured that models did not overtrain on noise or irrelevant patterns in the data. Additionally, loss curves provided a visual representation of model convergence, enabling researchers to assess training progress and identify potential issues such as underfitting or overfitting. The study further highlighted how hyperparameter tuning—such as adjusting learning rates or neuron counts—can significantly impact model performance.

Overall, this research demonstrates that custom-built neural network libraries can achieve competitive results in both theoretical benchmarks like XOR and real-world

applications like taxi trip duration prediction. By integrating robust methodologies for data preprocessing, feature engineering, model selection, and optimization, such libraries offer a flexible framework for addressing diverse machine learning challenges. Future work could explore extending this library with advanced techniques such as Neural Architecture Search (NAS) or multimodal approaches to enhance scalability and applicability across domains.

# 5 Conclusion

This study successfully implemented a custom neural network library to address two distinct machine learning problems: XOR classification and taxi trip duration prediction. The results demonstrated that neural networks are highly effective tools for solving both theoretical benchmarks and practical challenges when designed with appropriate architectures and methodologies. For the XOR problem—a classic test of non-linear separability—the network achieved near-perfect accuracy by leveraging activation functions like Sigmoid and Tanh to introduce non-linearity into its computations. This validated not only the library's design but also its robustness in handling fundamental machine learning tasks.

The taxi trip duration prediction task presented a more complex real-world scenario requiring extensive data preprocessing and feature engineering. Key features such as pickup/drop-off coordinates, trip distance, passenger count, and temporal variables were extracted from a large dataset comprising over one million samples. Visualization techniques provided critical insights into data distribution patterns; for example, geographic clustering revealed urban dominance in taxi activity around Manhattan. These analyses guided decisions about feature selection while ensuring that models were trained on meaningful inputs rather than noise or irrelevant variables.

Three neural network architectures—small, medium, and large—were evaluated to determine their effectiveness in capturing relationships within the dataset. The small model exhibited rapid convergence but
struggled with generalization due to its limited capacity. Conversely, while the large model offered greater complexity through additional layers and neurons, it introduced redundancy without substantial performance improvements. The medium-sized model emerged as the optimal architecture by balancing complexity with computational efficiency; it achieved superior RMSLE scores compared to both smaller models and pre-existing benchmarks.

The study also highlighted several key methodological considerations that contributed to its success. Early stopping proved invaluable in preventing overfitting by halting training once validation loss ceased improving. Loss curves provided a visual representation of training progress and helped identify potential issues such as underfitting or overfitting early in the process. Furthermore, hyperparameter tuning played a critical role in optimizing model performance by adjusting parameters like learning rates or neuron counts based on validation feedback.

The broader implications of these findings extend beyond this specific case study into other domains where neural networks are increasingly applied—such as healthcare diagnostics, sports analytics, renewable energy optimization, and space exploration. By demonstrating how custom-built libraries can achieve competitive results through careful design and implementation strategies, this research provides a foundation for future advancements in machine learning applications across diverse fields.

In conclusion, this study underscores the adaptability and reliability of neural networks when applied to both theoretical benchmarks like XOR classification and real-world challenges like taxi trip duration prediction. By integrating robust preprocessing techniques with well-designed architectures and optimization strategies, custom libraries can deliver high-performance solutions tailored to specific problems. Future research should explore extending these capabilities through advanced methods like Neural Architecture Search (NAS) or multimodal approaches to

further enhance scalability and versatility across domains.

# 6 Recommendations

Based on the comprehensive evaluation of machine learning performance using Python for neural network methods in urban transportation, numerous algorithmic approaches remain promising for further investigation and application in this field. Future research should explore a broad range of machine learning algorithms and optimization techniques that have been successfully employed in mathematics, statistics, physics, and computer science, thereby fostering an interdisciplinary framework to address complex urban challenges. The current study, which compared linear and logistic regression with advanced neural network models, has demonstrated that while traditional regression techniques provide valuable baseline insights and interpretability, artificial neural networks (ANNs) excel in capturing nonlinear relationships, as evidenced by their superior accuracy in predicting taxi trip durations. Building on these findings, researchers are encouraged to investigate additional algorithms such as support vector machines (SVM), random forests, gradient boosting machines, and other ensemble methods to further enhance prediction performance. In parallel, optimization methods rooted in linear programming, stochastic optimization, and convex analysis offer promising avenues to refine training procedures and feature selection, ultimately leading to more robust and efficient models. Moreover, emerging deep learning techniques, including convolutional neural networks (CNNs), recurrent neural networks (RNNs), deep belief networks (DBNs), and deep reinforcement learning algorithms, have the potential to leverage large-scale data sets and intricate spatiotemporal patterns inherent in urban transportation systems. In this context, the integration of automated approaches such as Neural Architecture Search (NAS) and automated machine learning (autoML) will be critical to systematically compare diverse model structures and identify optimal
configurations. The synergistic application of
these methods not only enhances predictive
accuracy but also provides deeper insights into
feature importance and system dynamics.
Future studies should also consider hybrid
models that combine the interpretability of tree-
based or rule-based approaches with the
flexibility and powerful representation
capabilities of deep learning, thus achieving a
balance between explainability and

performance. Additionally, interdisciplinary
research that incorporates frameworks from
physics—such as computational dynamics and
network theory—can yield novel perspectives
on traffic flow and congestion patterns, while
graph theory can be utilized to analyze the
complex interactions within transportation
networks.

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