Merge pull request #72 from DiogoRibeiro7/feat/reserve_branche

Feat/reserve branche

Author: github-actions[bot]

_posts/2021-05-10-estimating_uncertainty_neural_networks_using_monte_carlo_dropout.md

@@ -76,15 +76,15 @@ Monte Carlo dropout works by approximating the posterior distribution of a model
 Let $f(y|x)$ be the softmax output of the neural network for class $y$ given input $x$. Monte Carlo dropout involves generating $T$ samples $\{ f_t(y|x) \}_{t=1}^{T}$ by running the network $T$ times with different dropout masks. From these samples, we can compute:
 
 - **Predictive mean**: 
-  \[
+  $$
   \mathbb{E}[f(y|x)] = \frac{1}{T} \sum_{t=1}^{T} f_t(y|x)
-  \]
+  $$
   This gives the average probability assigned to class $y$ across the $T$ stochastic forward passes.
 
 - **Predictive variance**: 
-  \[
+  $$
   \text{Var}[f(y|x)] = \frac{1}{T} \sum_{t=1}^{T} (f_t(y|x) - \mathbb{E}[f(y|x)])^2
-  \]
+  $$
   This measures the dispersion of the model’s predictions, giving us an indication of how much the predictions vary due to dropout.
 
 The predictive variance is particularly useful for identifying inputs where the model is uncertain about its prediction. Large variance indicates that the model's predictions are inconsistent across different dropout configurations, signaling uncertainty.
@@ -97,9 +97,9 @@ Once we have the predictive mean and variance, the next challenge is to distill
 
 One intuitive approach is to use the probability of the predicted class as the uncertainty score. Specifically, we can compute the maximum predicted probability across all classes:
 
-\[
+$$
 \text{Uncertainty Score} = 1 - \max_y \mathbb{E}[f(y|x)]
-\]
+$$
 
 This score measures the model's confidence in its most likely prediction. A high value for $\max_y \mathbb{E}[f(y|x)]$ indicates high confidence in the predicted class, while a lower value suggests greater uncertainty.
 
@@ -109,9 +109,9 @@ This method is simple and easy to implement, but it has some limitations. For ex
 
 A more nuanced approach is to compute the entropy of the predictive distribution:
 
-\[
+$$
 H(\mathbb{E}[f(\cdot|x)]) = - \sum_{y} \mathbb{E}[f(y|x)] \log \mathbb{E}[f(y|x)]
-\]
+$$
 
 Entropy measures the overall uncertainty in the distribution of predicted probabilities. If the model assigns most of the probability mass to a single class, the entropy will be low, indicating high confidence. Conversely, if the probabilities are more evenly spread across classes, the entropy will be higher, indicating greater uncertainty.
 
@@ -121,15 +121,15 @@ This method captures uncertainty more comprehensively than the maximum class pro
 
 Another method is to use the variance of the predicted probabilities as a measure of uncertainty. The variance for each class $y$ is computed as:
 
-\[
+$$
 \text{Var}[f(y|x)] = \frac{1}{T} \sum_{t=1}^{T} (f_t(y|x) - \mathbb{E}[f(y|x)])^2
-\]
+$$
 
 To obtain a single uncertainty score, we can aggregate the variances across all classes. One common approach is to compute the total variance:
 
-\[
+$$
 \text{Total Variance} = \sum_{y} \text{Var}[f(y|x)]
-\]
+$$
 
 This score reflects the overall uncertainty in the model's predictions. High variance indicates that the model's predictions are inconsistent across different dropout configurations, suggesting that the model is unsure about its prediction.
 
@@ -139,23 +139,23 @@ Variance-based methods are particularly useful when the goal is to detect out-of
 
 In some cases, particularly when dealing with binary or reduced two-class problems, it may be useful to approximate the predictive distribution using a normal distribution. Specifically, we can model the output probabilities for class $y$ as a Gaussian distribution:
 
-\[
+$$
 p(y|x) \sim \mathcal{N}(\mu_y, \sigma_y^2)
-\]
+$$
 where $\mu_y = \mathbb{E}[f(y|x)]$ is the predictive mean and $\sigma_y^2 = \text{Var}[f(y|x)]$ is the predictive variance.
 
 For a two-class classifier, let $y$ be the predicted class (i.e., $y = \arg\max_y \mathbb{E}[f(y|x)]$) and $\neg y$ be the other class. The probability that a future evaluation of the classifier will also output $y$ is given by:
 
-\[
+$$
 u = \Pr[X \geq 0]
-\]
+$$
 where $X \sim \mathcal{N}(\mu_y - \mu_{\neg y}, \sigma_y^2 + \sigma_{\neg y}^2)$.
 
 This probability can be estimated using the error function:
 
-\[
+$$
 u = \frac{1}{2} \left[1 + \text{erf}\left(\frac{\mu_y - \mu_{\neg y}}{\sqrt{2 (\sigma_y^2 + \sigma_{\neg y}^2)}}\right)\right]
-\]
+$$
 
 This approach is particularly useful for binary classification problems or situations where multi-class problems can be reduced to a binary decision (e.g., when comparing the predicted class to all other classes). It provides a probabilistic estimate of the model’s confidence that a future evaluation will yield the same prediction.
 

_posts/2021-05-12-understanding_heart_rate_variability_through_lens_coefficient_variation_health_monitoring.md

@@ -0,0 +1,135 @@
+---
+author_profile: false
+categories:
+- Health Monitoring
+- Cardiovascular Health
+classes: wide
+date: '2021-05-12'
+excerpt: Discover the significance of heart rate variability (HRV) and how the coefficient of variation (CV) provides a more nuanced view of cardiovascular health.
+header:
+  image: /assets/images/data_science_2.jpg
+  og_image: /assets/images/data_science_2.jpg
+  overlay_image: /assets/images/data_science_2.jpg
+  show_overlay_excerpt: false
+  teaser: /assets/images/data_science_2.jpg
+  twitter_image: /assets/images/data_science_2.jpg
+keywords:
+- Heart Rate Variability
+- Coefficient of Variation
+- Cardiovascular Health
+- Fitness Monitoring
+- Stress Assessment
+seo_description: Explore how the coefficient of variation offers deeper insights into heart rate variability and health monitoring.
+seo_title: Understanding HRV and Coefficient of Variation
+seo_type: article
+summary: This article delves into heart rate variability (HRV), focusing on the coefficient of variation (CV) as a critical metric for understanding cardiovascular health and overall well-being.
+tags:
+- Heart Rate Variability
+- Coefficient of Variation
+- Health Metrics
+title: Understanding Heart Rate Variability Through the Lens of the Coefficient of Variation in Health Monitoring
+---
+
+Heart rate variability (HRV) is one of the most important indicators of cardiovascular health and overall well-being. It reflects the body’s ability to adapt to stress, rest, exercise, and environmental stimuli. Traditionally, HRV has been measured using several statistical tools, including standard deviation, root mean square of successive differences (RMSSD), and the low-frequency to high-frequency (LF/HF) ratio, to name a few.
+
+In the context of health monitoring, heart rate is frequently measured in beats per minute (bpm), and its variability can be crucial for understanding individual health dynamics. However, using only the standard deviation (SD) to evaluate HRV without considering the average heart rate can lead to an incomplete understanding. This is where the coefficient of variation (CV) becomes useful.
+
+The CV offers a deeper understanding of HRV by normalizing the standard deviation in relation to the mean heart rate, providing insight into how consistent or variable the heart rate is across different scenarios and conditions. This article will explore the relevance of CV in health, focusing on its application in heart rate monitoring and broader physiological contexts.
+
+## What is Heart Rate Variability (HRV)?
+
+### Definition of HRV
+
+HRV refers to the time variation between consecutive heartbeats. Unlike heart rate, which measures how many times the heart beats per minute, HRV focuses on the slight variations in the time intervals between individual beats. This variability is regulated by the autonomic nervous system (ANS), specifically the sympathetic and parasympathetic branches.
+
+- **Sympathetic nervous system**: Responsible for the "fight or flight" response, increasing heart rate and reducing HRV during stressful situations.
+- **Parasympathetic nervous system**: Responsible for the "rest and digest" functions, promoting relaxation and recovery, increasing HRV.
+
+### Importance of HRV in Health
+
+HRV is a non-invasive measure used in assessing the adaptability of the cardiovascular system. It has gained popularity in medical research and health monitoring because it reflects a person’s general state of health, fitness, stress levels, and recovery ability. High HRV is generally associated with good cardiovascular health, stress resilience, and efficient autonomic nervous system functioning. Conversely, low HRV is linked to stress, fatigue, and an increased risk of cardiovascular disease.
+
+## Traditional Methods of Measuring HRV
+
+### Standard Deviation (SD)
+
+One of the simplest and most common methods of calculating HRV is by measuring the standard deviation of the intervals between heartbeats, also known as the inter-beat intervals (IBIs). The standard deviation gives an idea of how much variation there is in the timing of the heartbeats.
+
+While useful, the SD alone doesn't account for differences in average heart rate between individuals or even within the same individual across different times or conditions. For example, two individuals could have similar SDs in their HRV, but if one person has a significantly higher average heart rate, the variability could represent a different degree of physiological stress or health condition.
+
+### Root Mean Square of Successive Differences (RMSSD)
+
+RMSSD is another commonly used metric for HRV and focuses on the short-term variability in heart rate, particularly useful for measuring the parasympathetic influence on heart rate. However, like SD, it also doesn’t account for the mean heart rate, and its interpretation might miss important contextual information.
+
+## Introducing the Coefficient of Variation (CV) in Heart Rate Analysis
+
+### Definition of Coefficient of Variation (CV)
+
+The coefficient of variation (CV) is a statistical measure that is often used to quantify the level of dispersion or variability of a data set relative to its mean. It is calculated as:
+
+$$
+CV = \frac{\text{Standard Deviation (SD)}}{\text{Mean (μ)}}
+$$
+
+By expressing variability in relation to the mean, CV allows for a normalized comparison across different contexts. In heart rate monitoring, this means understanding how variable someone’s heart rate is, given their average heart rate.
+
+### Advantages of CV in Health Monitoring
+
+- **Contextual Understanding of Variability**: CV allows us to compare variability in heart rates between individuals or within the same individual under different circumstances while accounting for differences in the mean heart rate.
+- **Adaptation to Various Conditions**: By using CV, one can assess how an individual's heart rate changes during different physiological states (e.g., rest, exercise, sleep). The CV offers insights into how adaptable the heart rate is in response to stressors or recovery periods.
+- **Improved Risk Assessment**: In clinical settings, CV can provide more nuanced risk assessments for heart conditions. Individuals with a high CV in their resting heart rate may indicate poor autonomic regulation or increased cardiovascular risk, while a low CV might indicate a more stable cardiovascular system.
+
+## Application of CV in Health Contexts
+
+### 1. Monitoring Fitness and Endurance
+
+One of the key applications of HRV and CV in health is in tracking physical fitness and endurance. Athletes often monitor their heart rates closely to ensure that they are training at an optimal level and recovering adequately.
+
+- **CV and Exercise Adaptation**: During exercise, an increase in heart rate is expected. However, understanding how heart rate fluctuates around the mean during intense physical activity or recovery can provide insights into an athlete's cardiovascular fitness. A higher CV might indicate a body struggling to maintain consistent performance under physical stress, whereas a lower CV could reflect better cardiovascular conditioning.
+- **CV as an Indicator of Overtraining**: Overtraining syndrome (OTS) can be detected using HRV and CV. When the CV increases significantly over time, it could signal an inability of the cardiovascular system to recover from continuous physical stress, highlighting the need for rest and recovery.
+
+### 2. Stress and Mental Health Assessment
+
+Heart rate variability is an established marker for emotional and psychological stress. The coefficient of variation can help distinguish between different states of stress and how the body copes with emotional or mental strain.
+
+- **CV and Chronic Stress**: Individuals under chronic stress typically exhibit reduced HRV. However, the CV can highlight how much variability exists relative to their resting heart rate. For example, a person with chronic stress may show a high CV, indicating fluctuations in heart rate as the body struggles to maintain equilibrium.
+- **Mental Health Conditions**: Certain mental health conditions such as anxiety, depression, and PTSD are associated with abnormal HRV patterns. Monitoring CV in these conditions can give healthcare providers additional insights into how well a person is responding to treatments such as therapy or medication.
+
+### 3. Cardiovascular Disease Risk and Recovery
+
+Monitoring heart rate variability and its associated metrics like CV has become a common practice in patients with cardiovascular disease (CVD). Reduced HRV has been associated with an increased risk of heart attacks and other cardiovascular events.
+
+- **CV in Predicting Cardiovascular Events**: CV can be a useful predictor in determining the stability of heart rate in patients with a history of cardiovascular disease. A higher CV may indicate a less stable heart rate and poorer autonomic regulation, signaling a higher risk of adverse events.
+- **Post-Surgery Monitoring**: Patients recovering from cardiovascular surgery or other invasive procedures often undergo continuous heart rate monitoring. Using CV during recovery can provide insights into how well the body is adjusting to post-surgical stress and whether the patient is at risk of complications like arrhythmias or ischemia.
+
+### 4. Sleep and Recovery Analysis
+
+Sleep plays a critical role in recovery and overall well-being, and HRV is closely tied to sleep quality. By analyzing the CV during sleep, particularly during REM and deep sleep phases, healthcare professionals and researchers can gain insight into how well the body is recovering.
+
+- **CV in Sleep Stages**: Different stages of sleep are associated with different heart rate patterns. By measuring the CV across these stages, one can determine how consistent or variable the heart rate is. A high CV during what should be a restful period could indicate poor sleep quality or disrupted recovery.
+- **Sleep Disorders**: Conditions such as sleep apnea or insomnia are known to disrupt HRV patterns. Tracking the CV during sleep in individuals with these disorders can help diagnose the severity of the condition and monitor the effectiveness of treatments like CPAP therapy or behavioral interventions.
+
+## Limitations of Using CV in Health Monitoring
+
+While the CV provides a valuable measure of heart rate variability relative to the mean, it is not without its limitations. Some of these include:
+
+- **Sensitivity to Mean Changes**: Because the CV is normalized by the mean, it can be overly sensitive to small fluctuations in the mean heart rate. For individuals with naturally low heart rates, even minor changes in HRV can result in large shifts in CV, which could be misleading.
+- **Lack of Temporal Information**: CV provides a summary measure of variability but does not account for when or how variability occurs. For example, a person might have a low CV over a 24-hour period but experience short, dangerous spikes in variability during certain times of the day.
+- **Not a Standalone Measure**: CV should be used alongside other HRV measures to get a full picture of heart rate dynamics. Other methods, such as the LF/HF ratio or RMSSD, may provide additional information about the underlying physiological processes at play.
+
+## Beyond CV: Other Relevant Measures in Heart Rate Monitoring
+
+While CV offers a valuable perspective on HRV, it is important to acknowledge other methods that can provide complementary information:
+
+- **Standard Deviation of NN intervals (SDNN)**: This is a time-domain method that measures the standard deviation of all NN intervals (the time between normal heartbeats) and is considered one of the most reliable global indicators of HRV.
+- **Root Mean Square of Successive Differences (RMSSD)**: RMSSD focuses on short-term variability and parasympathetic nervous system activity. It is often used in fitness and recovery tracking.
+- **Low Frequency to High Frequency Ratio (LF/HF)**: This ratio provides insight into the balance between sympathetic and parasympathetic activity. A higher LF/HF ratio generally indicates higher sympathetic activity, which is linked to stress or strain on the body.
+- **pNN50**: This measures the percentage of successive NN intervals that differ by more than 50ms. It is a commonly used marker of parasympathetic nervous system activity.
+
+## The Value of Coefficient of Variation in Heart Rate Monitoring
+
+The coefficient of variation (CV) offers a unique and insightful way to understand heart rate variability in the context of health. By normalizing variability with respect to the mean heart rate, CV provides a clearer picture of how consistent or unstable a person’s heart rate is under different conditions.
+
+In the fields of fitness, stress monitoring, cardiovascular disease, and sleep analysis, CV can offer additional insights that complement traditional HRV metrics. When used alongside other measures such as RMSSD, SDNN, and LF/HF, CV becomes a powerful tool for assessing overall health, guiding recovery, and providing early warnings for potential health risks.
+
+As healthcare continues to evolve with the integration of wearable technology and real-time monitoring systems, CV's role in personalized health management is likely to expand, offering both patients and providers more nuanced and actionable data.

markdown_file_processor.py

@@ -4,6 +4,7 @@
 # List of stop words to remove from file names
 STOP_WORDS = {"and", "or", "the", "a", "an", "that", "this", "is", "in", "on", "with", "of", "for", "to"}
 
+
 def rename_markdown_file(file_path: str):
     """
     Renames the markdown file so that the name part after the date is in lowercase,