In the vast ocean of the digital world, the data scientist serves us both as a navigator and interpreter. The chart courses through the tumultuous sea of information that surrounds us every day and only seems to be growing and deriving insights from the ebb and flow of data.
But, before our heroes embark on this journey, they need to learn the tools of the trade – basically, statistical concepts. These concepts form the bedrock of data science:
Probability Theory – We start at the very beginning of the statistical concepts – Probability Theory- This branch of mathematics deals with the analysis of random phenomena, providing us with the tools to quantify uncertainty and make informed decisions in the face of incomplete information
Sample Spaces and Events – At the heart of the probability theory lies the concept of the sample space (Ω), which encompasses all the possible outcomes or random processes. Each individual outcome, in this situation, is known as an elementary event, while a subset of the sample space is referred to as an event.
Axioms of Probability - The foundation of probability rests upon three fundamental axioms”
These axioms, like thermodynamics, govern the behavior of probabilities in our universe.
Conditional Probability and Bayes’ Theorem – Largely considered one of the most elements in Data Science, Bayes’ Theorem. Conditional Probability is the likelihood of an event occurring provided that another event has already occurred. This can be expressed mathematically as :
P(A|B) = P(A ∩ B) / P(B)
This seemingly simple formula gives rise to one of the most powerful tools in the hands of a Data Scientist: Bayes Theorem. As explained before, Bayes’ Theorem is considered one of the most important elements in Data Science. This theorem allows us to update our understanding of belief in the face of new evidence:
P(A|B) = P(B|A) * P(A) / P(B)
For laymen, Bayes’ Theorem is the statistical equivalent of a shapeshifter, adapting to our understanding of probabilities as new data comes to light.
PROBABILITY DISTRIBUTION
We have, so far, laid the foundation for working with the probability theory, we can now turn our attention to the various shapes that data can assume. Probability distribution, in essence, is the mathematical functions that describe the likelihood of different outcomes in a random statement.
DISCRETE DISTRIBUTIONS
P(X = k) = p^k * (1-p)^(1-k), for k ∈ {0, 1}
Where P is the probability of success. It can be compared to the quantum state of Schrödinger's cat.
P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
Where n is the number of trials, k is the number of successes and p is the probability of success on each trial. (C(n,k)) represents the binomial coefficient, also known as “n choose k”
CONTINUOUS DISTRIBUTIONS
Normal Distribution – Also known as Gaussian Distribution, this bell- shaped curve is ubiquitous in nature and statistics. The probability density function of a normal distribution is given by:
f(x) = (1 / (σ√(2π))) * e^(-(x-μ)^2 / (2σ^2))
Where µ is the mean and σ is the standard deviation. The normal distribution is the statistical embodiments of all things. It is the distribution of heights in a population, of errors in measurements, and many others.
Exponential Distribution – The exponential distribution describes the time between events in a Poisson point process, making it particularly useful in modeling the time until the time until an event happens.
An exponential density function is :
f(x) = λe^(-λx), for x ≥ 0
Where λ is the rate parameter.
INFERENTIAL STATISTICS: FROM SAMPLE TO POPULATION
Now that we have made ourselves familiar with the tools of probability theory and an understanding of various distributions, let us move into inferential statistics. Inferential statistics seeks to conclude populations based on samples.
Point Estimation – Point Estimation is the process of using sample data to calculate a single value (a point estimate, as the name suggests), which serves as more of a “best guess estimate” of an unknown population parameter. Common point estimators include the sample mean (x̄) for the population mean (μ) and the sample variance (s^2) for the population variance (σ^2). However, it is important to note here that a point estimate without a measure of its precision can be dangerously directionless.
Variance – Variance is a measure of variability in a dataset that quantifies how far individual numbers are from the mean (average) value. It's calculated by taking the average of the squared differences from the mean. Variance is crucial in statistics as it helps describe the distribution of data and is used in many statistical tests and models. It's denoted by σ² for population variance and s² for sample variance. The square root of variance gives us the standard deviation, another important measure of spread in statistics.
Interval Estimation – Interval estimation addresses this limitation by providing a range of values likely to contain the population parameters. The most common form is the confidence interval. For a normally distributed population with known variance, the (1-a)100% confidence interval for the population mean is given by :
(x̄ - z_(α/2) * (σ/√n), x̄ + z_(α/2) * (σ/√n))
Where z_(α/2) is the critical value from the standard normal distribution.
HYPOTHESIS TESTING–Hypothesis testing is the stage where all assumptions are put on trial and act as judge, jury, and executioner. This process involves formulating two competing hypotheses against each other, explained below:
Null Hypothesis: The status quo – The assumption of no effect or no difference.
Alternative Hypothesis: In this case we call this the challenger, positing a significant change or difference.
How it Works for a Data Scientist
The Data Scientist goes through the following steps for hypothesis testing:
THE INFAMOUS P-VALUE
Largely considered the enfant terrible of inferential statistics is the P value. Formally, the P value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is correct.
In practice, however. The p-value has become a litmus test with scores below (0.05) which results are considered significant and scores above being ignored as insignificant. It is also critical to remember that the >.05 value that data scientists chase is merely a tool, and is not a truth serum. Counting on it entirely is at the peril of the data scientist.
REGRESSION ANALYSIS – Relationships in Data
With Regression Analysis, the point Is to understand and quantify the relationships between variables, to predict the future based on the past, and to explain any variance in data with a mathematical equation.
Simple Linear Regression – Simple linear regression is the statistical equivalent of drawing a straight line through a cloud of points, to capture the relationships between them. The equation is:
Y = β₀ + β₁X + ε
Where Y is the dependent variable, X is the independent variable, , β₀ is the y-intercept, β₁ is the slope, and ε is the error term.
Multiple Linear Regression – At times, simple linear regression proves to be insufficient to capture the complexity of the data, and multiple linear regression comes into play. This technique allows us to model the relationship between a variable and multiple dependent variables.
The equation for multiple linear regression extends our previous model to:
Y = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε
Where X₁, X₂, ..., Xₖ are the independent variables, and β₁, β₂, ..., βₖ are their respective coefficients.
LOGISTIC REGRESSION
When a dependent variable is categorical rather than continuous, logistic regression comes to the rescue. This technique models the probability of an instance belonging to a particular class. The name of the original logistic function is
P(Y=1) = 1 / (1 + e^(-z))
Where z = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ
ASSUMPTIONS AND DIAGNOSIS
Following in the steps of any good statistical technique, regression analysis comes with a set of assumptions that must be met for our result to be valid. In order of importance, these are:
These assumptions form a solid ground for regression and should not be ignored under any criteria. To check them, there are several diagnostic tools:
DIMENSIONALITY REDUCTION
As we delve deeper into data complexities, we are often faced with datasets of high dimensionality. While more features provide more information, they also bring the risk of dimensionality – a phenomenon where the volume of the space increases so rapidly that the available data becomes sparse.
Principal Component Analysis – PCA is a statistical procedure that uses an orthogonal transformation to convert a set of values of linearly uncorrelated variables into a set of linearly uncorrelated variables called Principal Components.
The Steps involved in PCA are:
t-Distributed Stochastic Neighbor Embedding (t-SNE)
When even PCA proves insufficient (they rarely do, but they sometimes do prove insufficient) for visualizing, t-SNE is the savior we turn to. The t-SNE is particularly adept at visualizing high- dimensional data by giving each data point a location in a two- or three- dimensions map. T-SNE works by minimizing the divergence between two distributions. Things to note: a) t-SNE is a distribution that measures pairwise similarities of the input objects, and b) it is a distribution that measures pairwise similarities of the corresponding low-dimensional points in the embedding.
CONCLUSION
As we conclude our exploration of essential statistical concepts for beginners, it is critical to understand that mastering these fundamentals is just your entry into the realm of data science. In the rapidly changing technological landscape we see around us; globally renowned certifications have become indispensable for career advancement. These certifications act as powerful validations of your expertise, opening doors to lucrative opportunities. For aspiring data scientists and seasoned professionals alike, obtaining a respected and renowned professional certification can lead to higher salaries, and even increase job security. Organizations like the Data Science Institute (USDSI.ORG) offer compelling certification options. With rigorous curricula that align directly with industry demands. So go forth, get certified, and show employers your readiness for data science programming challenges.
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