Month: September 2023

• Does the Number of Folds Matter in Cross Validation?

K-Fold Cross Validation is an essential technique in machine learning, particularly when the performance of a model shows significant variance based on the train-test split. While many practitioners often use 5 or 10 folds, there is no strict rule or norm dictating these numbers. In essence, one can use as many folds as deemed appropriate for the specific dataset and problem at hand.

• Experimentation with Different Fold Numbers:

I conducted an experiment using 5-fold cross validation and obtained an R-Squared value of 34%.

To delve deeper into the behavior of the model, I aim to experiment with varying numbers of folds. The objective is to observe any changes in the R-Squared value as the number of folds changes.

Beyond this, I intend to take unequal sets for training and testing, then perform regression to see the resulting R-Squared values. By doing so, I hope to gain a deeper understanding of how the model behaves under different conditions.

In Continuation to my previous post, I performed multi-linear regression using three datasets: %diabetes, %inactivity, and %obesity. From my analysis, I have identified 354 common count codes present across all three sheets. The results of our regression yielded a standard error of 0.59, a multiple R-value of 0.58, and an R-squared value of 0.34. These metrics suggest that our model might not be highly reliable or effective in explaining the observed outcomes.

Upon closer inspection of the data, we noticed an anomaly. Certain values, which aren’t duplicates, were treated as such in the regression model. To address this, I am considering implementing cross-validation. My plan is to label each of these “duplicate” data points uniquely, ensuring that our model recognizes them as distinct variables.

Later, to enhance the robustness of our model, I will be employing k-fold validation. I will divide our dataset into five segments. In this approach, 1/5th of the data will serve as the training set, while the model will be tested against the remaining 4/5ths. By rotating the training set and averaging the results from these five models, I aim to obtain a more accurate estimate of the test error.

Update regarding project 1:

Today, I have conducted multiple linear regression analysis on the provided data, specifically focusing on variables common to all sheets, namely “%diabetes,” “%inactivity,” and “%obesity.” The results of this analysis are summarized as follows:

• Multiple R: 0.583729108
• R Square: 0.340739671
• Adjusted R Square: 0.336983202
• Standard Error: 0.593139754
• Observations: 354

The multiple R value indicates the correlation between the independent variables and the dependent variable. The R Square value represents the proportion of variance in the dependent variable that can be explained by the independent variables. The Adjusted R Square value adjusts the R Square for the number of predictors. The Standard Error provides an estimate of the standard deviation of the errors in the regression model, and the number of Observations reflects the sample size used in the analysis.

I watched a video today about cross-validation and bootstrap. I learned that we can estimate a model’s test error using the training error as a rough indicator. Typically, the test error is higher than the training error because the model faces unseen data during testing. To refine this estimate, we can apply methods like the cp statistic, AIC, and BIC, which adjust the training error mathematically to better reflect the test error.

The video also introduced the Validation Set Approach. It involves splitting the data into two parts: the training set and the validation set. The model is trained on the training set, and then we use this trained model to predict outcomes for the validation set. The resulting validation set error gives us an estimate of the test error.

However, there are some downsides to this approach. The validation set error can vary significantly depending on how we randomly split the data, making it less stable. Additionally, since we only train the model on a subset of the data (the training set), it might not capture the full complexity and diversity of the dataset. This can lead to an overestimate of the test error when we eventually fit the model to the entire dataset.

In summary, while the Validation Set Approach is a useful way to estimate test error, it has limitations due to variability and potential model underfitting. Care should be taken when interpreting its results, especially when applying the model to the entire dataset.

In our recent class, we learnt about the Crab Molt Model, a potent linear modeling approach designed for situations where two variables exhibit non-normal distribution, skewness, high variance, and high kurtosis. The central aim of this model is to predict pre-molt size based on post-molt size.

We learnt the concept of statistical significance, particularly focusing on differences in means. Using data from “Stat Labs: Mathematical Statistics Through Applications,” Chapter 7, page 139, we constructed a model and generated a linear plot. While plotting graphs for post-molt and pre-molt sizes, we observed a notable difference in means. Intriguingly, the size and shape of these graphs bore a striking similarity, differing by just 14.68 units.

To assess the statistical significance of this observed difference, we initially considered utilizing the common t-test, typically used for comparing means in two-group scenarios. However, our project introduced a complexity: it involved three variables, rendering the t-test inappropriate for our analysis.

The Crab Molt Model and the exploration of mean differences by the t-test, are the tools for deciphering data complexities. Nonetheless, in the face of intricate, multi-variable scenarios, embracing advanced statistical methodologies becomes crucial for uncovering meaningful insights and advancing our understanding of statistical significance.

The t-test is not applicable in our Project 1, which involves three variables, as it is designed for comparisons between two variables. Instead, we need to explore advanced techniques like ANOVA or regression analysis to assess the significance of differences in means in our complex scenario.

Linear regression is a powerful tool in data analysis, but it relies on some crucial assumptions. One of these is homoskedasticity, which means the variance of errors should be constant across different levels of independent variables. If this assumption doesn’t hold, our regression results may not be reliable. This is where Python’s Breusch-Pagan test, available in the statsmodels library.

To detect Heteroskedasticity with the Breusch-Pagan Test in python I used the following steps:

1. 1. Import the necessary libraries, including statsmodels.
   2. Fit an Ordinary Least Squares (OLS) regression model to your data, specifying the dependent and independent variables.
   3. Use the het_breuschpagan function from statsmodels to perform the Breusch-Pagan test on the residuals of the regression model.
   4. The p-value obtained from the Breusch-Pagan test is crucial for identifying heteroskedasticity. If this p-value is below a chosen significance level, typically 0.05, it suggests that heteroskedasticity may be affecting the reliability of your regression analysis.

Linear 3D models:

Whenever we must examinate the relationship between 3 variables in our case %diabetes, %inactivity and %obesity. We can use this model to visualize and understand the intricate relationships among three variables in a three-dimensional space. Variables don’t always act independently. Sometimes, one variable’s effect on the outcome depends on the value of another variable. These interactions can significantly influence your model’s predictions and are crucial to consider.

In data analysis, the first step is to ensure the collection of clear and accurate data. We obtained data on %diabetics, %inactivity, and %obesity from a project sheet and employed Python’s NumPy for essential statistical computations, such as medians, means, and standard deviations. These calculations provide us with a foundational understanding of the dataset.

Our primary objective was to unveil the relationship between the percentage of diabetics (%diabetics) and the percentage of inactive individuals (%inactivity). To achieve this, we constructed a scatterplot representing each region as a data point. This visual aid played a crucial role in assessing the connection between these two variables. Subsequently, we utilized the scatterplot to compute the R-squared value, a metric that quantifies the strength of this relationship. A higher R-squared value signifies a more robust connection, potentially shedding light on the significant contribution of inactivity to diabetes rates. We also meticulously examined residuals to validate our model and ensure the absence of anomalies or outliers.

Furthermore, through the inclusion of histograms and density plots. These visualizations provided valuable insights into how the data was distributed across our dataset. With these powerful analytical tools, we aimed to gain a comprehensive understanding of the intricate relationship between the percentage of diabetics and the percentage of inactive individuals. In essence, our systematic approach encompassed precise data collection, thorough statistical analysis using NumPy, and insightful visualizations, all contributing to unraveling the connection between %diabetics and %inactivity and enhancing our comprehension of diabetes rates.

Have you ever wondered how scientists and researchers determine if the patterns they find in data are real or just random chance? Enter the P-value, a nifty statistical tool that helps us separate the signal from the noise in data analysis.

P-value, short for “probability value,” is a number that tells us the likelihood of something happening by chance. Imagine you are flipping a coin, and you suspect it is rigged to land on heads more often. The P-value helps us figure out if the evidence supports your suspicion or if the results could easily occur randomly.

So, why is P-value important? It helps us decide whether the patterns we see in data are likely due to a real cause or just chance. A small P-value, usually less than 0.05, suggests that our findings are probably not random. This gives us confidence that we are onto something meaningful.

In essence, the P-value is your data analysis sidekick. It tells you if your findings are worth getting excited about or if they could just be a lucky fluke. Remember, though, while a low P-value is a good sign, it is not the only thing to consider in data analysis. Always look at the bigger picture, and use P-values wisely to unlock the secrets hidden within your data.

Today’s class was about simple linear regression, a technique for understanding relationships between variables. Our key takeaway was the importance of collecting descriptive data to make the analysis process smoother. Once we have gathered this information, we can use it to create informative graphs, and these graphs help us calculate crucial statistics like the median, mean, standard deviation, skewness, and kurtosis.

To put this into practice, I took the data provided in our project sheet and organized it into a new sheet, focusing on %diabetics, %inactivity, and %obesity. I then calculated how the percentage of diabetics relates to both the percentage of inactivity and the percentage of obesity, individually. I used a Python library called NumPy to compute key statistics such as the median, mean, standard deviation, skewness, and kurtosis.

Next on the agenda is creating graphs using these statistics. Specifically, I will be looking at the correlation between %diabetes and %inactivity, visualizing this relationship with a scatterplot. This scatterplot will also help me determine the R-squared value, which is vital in gauging the strength of the connection between these two variables. After that, I’ll dive into analyzing the residuals to gain deeper insights into how well our linear model is performing and whether it’s a valid representation of the data.

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