Top Mathematics discussions
NishMath - #Prediction
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@phys.org
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phys.org
A new mathematical model developed by the University of Rovira i Virgili's SeesLab research group, along with researchers from Northeastern University and the University of Pennsylvania, has made it possible to predict human mobility between cities with high precision. The model offers a simpler and more efficient way than current systems and is a valuable tool for understanding how people move in different contexts, which is crucial for transport planning, migration studies, and epidemiology. The research was published in the journal *Nature Communications*.
The model builds on traditional "gravitational models," which estimate mobility based on population size and distance between cities. While these models are simple, they lack accuracy. Modern approaches leverage artificial intelligence and machine learning to incorporate many variables besides origin and destination, such as the density of restaurants and schools, and the socio-demographic characteristics of the population. The COVID-19 pandemic highlighted the importance of predicting mobility for understanding the spread and evolution of viruses. Recommended read:
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Will@math.stackexchange.com
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math.stackexchange.com
, medium.com
A recent analysis has delved into the probabilistic interpretation of linear regression coefficients, highlighting the differences in reasoning when using expected values versus covariances. It has been shown that when calculating regression coefficients, employing expected values leads to correct formulations that correspond to the ordinary least squares (OLS) method. Specifically, the formula a=E[XY]/E[X^2] is derived using the expected value of the product of the independent and dependent variables. This approach aligns with the traditional understanding of linear regression where a model is expressed as Y=aX+ε, with ε being a centered error term independent of X.
However, using covariances for the probabilistic interpretation fails, especially in models without an intercept term. While covariance is often used to calculate the correlation between variables, the derived formula a=cov(X,Y)/var(X) does not align with the correct regression coefficient when there isn't an intercept. This divergence arises because the assumption of an intercept is implicit when using covariance, and its absence invalidates the formula using covariance. The study clarifies how formulas are derived in both scenarios and why the probabilistic reasoning fails when using covariances in situations where there is no intercept included in the model. The use of empirical means versus population means was also discussed to explore the nuances further. Recommended read:
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Amir Najmi@unofficialgoogledatascience.com
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Data scientists and statisticians are continuously exploring methods to refine data analysis and modeling. A recent blog post from Google details a project focused on quantifying the statistical skills necessary for data scientists within their organization, aiming to clarify job descriptions and address ambiguities in assessing practical data science abilities. The authors, David Mease and Amir Najmi, leveraged their extensive experience conducting over 600 interviews at Google to identify crucial statistical expertise required for the "Data Scientist - Research" role.
Statistical testing remains a cornerstone of data analysis, guiding analysts in transforming raw numbers into actionable insights. One must also keep in mind bias-variance tradeoff and how to choose the right statistical test to ensure the validity of analyses. These tools are critical for both traditional statistical roles and the evolving field of AI/ML, where responsible practices are paramount, as highlighted in discussions about the relevance of statistical controversies to ethical AI/ML development at an AI ethics conference on March 8. Recommended read:
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@vatsalkumar.medium.com
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medium.com
Recent articles have focused on the practical applications of random variables in both statistics and machine learning. One key area of interest is the use of continuous random variables, which unlike discrete variables can take on any value within a specified interval. These variables are essential when measuring things like time, height, or weight, where values exist on a continuous spectrum, rather than being limited to distinct, countable values. The concept of the probability density function (PDF) helps us to understand the relative likelihood of a variable taking on a particular value within its range.
Another significant tool being explored is the binomial distribution, which can be applied using programs like Microsoft Excel to predict sales success. This distribution is suited to situations where each trial has only two outcomes – success or failure, like a sales call resulting in a deal or not. Using Excel, one can calculate the probability of various sales outcomes based on factors like the number of calls made and the historical success rate, aiding in setting achievable sales goals and comparing performance over time. Also, the differentiation between binomial and poisson distribution is critical for correct data modelling, with binomial experiments requiring fixed number of trials and two outcomes, unlike poisson. Finally, in the world of random variables, a sequence of them conditionally converging to a constant value has been discussed, highlighting that if the sequence converges, knowing it passes through some point doesn't change the final outcome. Recommended read:
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@ameer-saleem.medium.com
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Recent discussions and articles have highlighted the importance of linear regression as a foundational tool in statistical modeling and predictive analysis. This classic approach, while simple, remains a powerful technique for understanding relationships between variables, using both theoretical frameworks and practical demonstrations. The core concept of linear regression involves finding a best-fit line that helps predict a dependent variable based on one or more independent variables. This method is applicable across many fields for forecasting, estimation, and understanding the impact of factors within datasets.
Linear regression models, at their basic core, use equations to describe these relationships. For a simple linear regression with one independent variable, this is represented as Y = wX + b where Y is the predicted variable, X is the input variable, w is the weight, and b is the bias. In more complex models, multiple variables are taken into account with equations extended to Y = w1X1 + w2X2 + … + wnXn + b. Practical implementation often involves using programming languages like R, with packages that can easily produce regression models, statistical summaries, and visualizations for analysis, data preperation and exploration. Recommended read:
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@digitaltechneha.medium.com
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www.analyticsvidhya.com
, gsapra.medium.com
Probability and statistical methods are being explored across various fields, including applications of probability distributions with examples from finance and error analysis. The focus includes an examination of counting techniques in probability and the study of joint, marginal, and conditional probabilities. This research also delves into the transformation of distributions, all of which are crucial for real-world applications.
This area of study uses mathematical and computational methods like Monte Carlo simulations to estimate probabilities. The work also explores how data analysis has evolved from traditional statistical methods to AI-driven insights, along with the fundamentals of linear regression, which serves as a key tool in data analysis. Furthermore, the work considers methods for hypothesis testing such as one-sample, two-sample, and paired t-tests using real world examples. Another area being examined is descriptive measures of association, and data management techniques such as SQL server statistics. A specific challenge was also examined, that of finding integer tetrahedrons with a given volume. Recommended read:
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