YouTube Link: https://youtu.be/UoHu27xoTyc
** Machine Learning Engineer Masters Program: https://www.edureka.co/machine-learning-certification-training **
This Edureka PPT on Least Squares Regression Method will help you understand the math behind Regression Analysis and how it can be implemented using Python.
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It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
YouTube Link: https://youtu.be/UoHu27xoTyc
** Machine Learning Engineer Masters Program: https://www.edureka.co/machine-learning-certification-training **
This Edureka PPT on Least Squares Regression Method will help you understand the math behind Regression Analysis and how it can be implemented using Python.
Follow us to never miss an update in the future.
YouTube: https://www.youtube.com/user/edurekaIN
Instagram: https://www.instagram.com/edureka_learning/
Facebook: https://www.facebook.com/edurekaIN/
Twitter: https://twitter.com/edurekain
LinkedIn: https://www.linkedin.com/company/edureka
Castbox: https://castbox.fm/networks/505?country=in
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
This Chapter is part of previous published ch.1 and ch.3 and its use for undergraduate students in physics department. also, you can use it for mathematical and Statistical courses and for those experimental courses of data fitting.
This lecture covers Linear regression
Linear Regression is one of the most common, some 200 years old and most easily understandable in statistics and machine learning
it comes under predictive modelling.
Predictive modelling is a kind of modelling here the possible output(Y) for the given input(X) is predicted based on the previous data or values.
A widely used principle for fitting straight lines is the method of least squares by Gauss and Legendre
Find parameters for a model function that minimizes the error between values predicted by the model and those known from the training set
Applied Numerical Methods Curve Fitting: Least Squares Regression, InterpolationBrian Erandio
Correction with the misspelled langrange.
and credits to the owners of the pictures (Fantasmagoria01, eugene-kukulka, vooga, and etc.) . I do not own all of the pictures used as background sorry to those who aren't tagged.
The presentation contains topics from Applied Numerical Methods with MATHLAB for Engineers and Scientist 6th and International Edition.
A TRIANGLE-TRIANGLE INTERSECTION ALGORITHM csandit
The intersection between 3D objects plays a prominent role in spatial reasoning, geometric
modeling and computer vision. Detection of possible intersection between objects can be based
on the objects’ triangulated boundaries, leading to computing triangle-triangle intersection.
Traditionally there are separate algorithms for cross intersection and coplanar intersection.
There is no single algorithm that can intersect both types of triangles without resorting to
special cases. Herein we present a complete design and implementation of a single algorithm
independent of the type of intersection. Additionally, this algorithm first detects, then intersects
and classifies the intersections using barycentric coordinates. This work is directly applicable to
(1) Mobile Network Computing and Spatial Reasoning, and (2) CAD/CAM geometric modeling
where curves of intersection between a pair of surfaces is required for numerical control (NC)
machines. Three experiments of the algorithm implementation are presented as a proof this
feasibility.
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Empowering the Data Analytics Ecosystem: A Laser Focus on Value
The data analytics ecosystem thrives when every component functions at its peak, unlocking the true potential of data. Here's a laser focus on key areas for an empowered ecosystem:
1. Democratize Access, Not Data:
Granular Access Controls: Provide users with self-service tools tailored to their specific needs, preventing data overload and misuse.
Data Catalogs: Implement robust data catalogs for easy discovery and understanding of available data sources.
2. Foster Collaboration with Clear Roles:
Data Mesh Architecture: Break down data silos by creating a distributed data ownership model with clear ownership and responsibilities.
Collaborative Workspaces: Utilize interactive platforms where data scientists, analysts, and domain experts can work seamlessly together.
3. Leverage Advanced Analytics Strategically:
AI-powered Automation: Automate repetitive tasks like data cleaning and feature engineering, freeing up data talent for higher-level analysis.
Right-Tool Selection: Strategically choose the most effective advanced analytics techniques (e.g., AI, ML) based on specific business problems.
4. Prioritize Data Quality with Automation:
Automated Data Validation: Implement automated data quality checks to identify and rectify errors at the source, minimizing downstream issues.
Data Lineage Tracking: Track the flow of data throughout the ecosystem, ensuring transparency and facilitating root cause analysis for errors.
5. Cultivate a Data-Driven Mindset:
Metrics-Driven Performance Management: Align KPIs and performance metrics with data-driven insights to ensure actionable decision making.
Data Storytelling Workshops: Equip stakeholders with the skills to translate complex data findings into compelling narratives that drive action.
Benefits of a Precise Ecosystem:
Sharpened Focus: Precise access and clear roles ensure everyone works with the most relevant data, maximizing efficiency.
Actionable Insights: Strategic analytics and automated quality checks lead to more reliable and actionable data insights.
Continuous Improvement: Data-driven performance management fosters a culture of learning and continuous improvement.
Sustainable Growth: Empowered by data, organizations can make informed decisions to drive sustainable growth and innovation.
By focusing on these precise actions, organizations can create an empowered data analytics ecosystem that delivers real value by driving data-driven decisions and maximizing the return on their data investment.
2. What is Curve Fitting?
• Curve fitting is the process of constructing a curve, or mathematical functions, which possess
closest proximity to the series of data. By the curve fitting we can mathematically construct
the functional relationship between the observed fact and parameter values, etc. It is highly
effective in mathematical modelling some natural processes.
• It is a statistical technique use to drive coefficient values for equations that express the value
of one(dependent) variable as a function of another (independent variable).
3. Why Curve Fitting?
• The main purpose of curve fitting is to theoretically describe experimental data with a model
(function or equation) and to find the parameters associated with this model.
• Mechanistic models are specifically formulated to provide insight into a chemical, biological or
physical process that is thought to govern the phenomenon under study.
Parameters derived from mechanistic models are quantitative estimation of real system
properties (rate constants, dissociation constants, catalytic velocities etc.) .
• It is important to distinguish mechanistic models from empirical models that are mathematical
functions formulated to fit a particular curve but those parameters do not necessarily corresponds to
a biological, chemical or physical property.
4. There are two general approaches for curve fitting:
• Least squares regression:
Data exhibit a significant degree of scatter. The strategy is to derive a single curve that
represents the general trend of the data.
• Interpolation:
Given a set of data that results from an experiment (simulation based or otherwise), or
perhaps taken from a real-life physical scenario, we assume there is some function that
passes through the data points and perfectly represents the quantity of interest at all non-
data points. With interpolation we seek a function that allows us to approximate such that
functional values between the original data set values may be determined (estimated). The
interpolating function typically passes through the original data set.
5. Interpolation
• The simplest type of interpolation is linear interpolation, which simply connects each data
point with a straight line.
• The polynomial that links the data points together is of first degree, e.g., a straight line.
• Given data points f(c) and f(a), where c>a.
We wish to estimate f(b) where b∈ [𝑎 𝑐] using linear interpolation.
6. Contd…
• The linear interpolation function for functional values between a and c can be found using
similar triangles or by solving of system of two equations for two unknowns.
• The slope intercept form for a line is:
𝑦 = 𝑓 𝑥 = 𝛼𝑥 + 𝛽, 𝑥 𝜖 𝑎, 𝑐
As boundary conditions we have that this line must pass through the point pairs 𝑎, 𝑓 𝑎 and
𝑏, 𝑓 𝑏 .
Now using this we can calculate 𝛼 and 𝛽. By substituting the values of 𝛼 and 𝛽 we can form
the equation as:
𝑓 𝑏 = 𝑓 𝑎 +
𝑏 − 𝑎
𝑐 − 𝑎
[𝑓 𝑐 − 𝑓(𝑎)]
7. Contd…
• Suppose we have the following velocity versus time data (a car accelerating from a rest
position).
• Linear Interpolation result :
• Cubic Interpolation:
8. Linear Regression
• The Method of Least Squares is a procedure to determine the best fit line to data; the proof uses
simple calculus and linear algebra.
• The basic problem is to find the best fit straight line y = ax + b given that, for n ∈ {1, . . . , N}, the
pairs (𝑥 𝑛, 𝑦𝑛) are observed.
• Consider the distance between the data and points on the line.
• Add up the length of all the red and blue vertical lines.
• This is an expression of the ‘error’ between data and fitted line.
• The one line that provides a minimum error is then the ‘best’ straight line.
9. Contd…
• Least square regression.
With linear regression a linear equation, is chosen that fits the data points such that the sum of
the squared error between the data points and the line is minimized
The squared distance is computed with respect to the y – axis.
Given a set of data points
𝑥 𝑘, 𝑦 𝑘 , 𝑘 = 1, … , 𝑁
The mean squared error (mse) is defined as
𝑚𝑠𝑒 =
1
𝑁
𝐾=1
𝑁
[𝑦 𝑘 − 𝑦1 𝑘]2
=
1
𝑁
𝐾=1
𝑁
[𝑦 𝑘 − (𝑚𝑥 𝑘+𝑏)]2
The minimum mse is obtained for particular values of m and b. Using calculus we compute the
derivative of the mse with respect to both m and b.
1. derivative describes the slope
2. slope = zero is a minimum ==> take the derivative of the
10. Contd…
𝜕𝑒𝑟𝑟
𝜕𝑚
= −2
𝑖=1
𝑛
𝑥𝑖 𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏 = 0
𝜕𝑒𝑟𝑟
𝜕𝑏
= −2
𝑖=1
𝑛
𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏 = 0
Solve for m and b.
The resulting m and b values give us the best straight line (linear) fit to the data
11. For higher order polynomials.
• Polynomial curve fitting
• Consider the general form for a polynomial of order 𝑗
𝑓 𝑥 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥2 + ⋯ + 𝑎𝑗 𝑥 𝑗 = 𝑎0 +
𝑖=1
𝑗
𝑎 𝑘 𝑥 𝑘
• The curve that gives minimum error between data and the fit 𝑓(𝑥) is best.
• Quantify the error for these two second order curves.
• Add up the length of all the red and blue vertical lines.
• Pick curve with minimum total error
12. Contd…
Error Least Square approach.
• The general expression for any error using the least squares approach is
𝑒𝑟𝑟 = (𝑑𝑖)2= (𝑦1 − 𝑓(𝑥1))2+(𝑦2 − 𝑓(𝑥2))2+(𝑦3 − 𝑓(𝑥3))2+(𝑦4 − 𝑓(𝑥4))2
• Now minimizing the error
𝑒𝑟𝑟 =
𝑖=1
𝑛
(𝑦𝑖 − (𝑎0 + 𝑎1 𝑥𝑖 + 𝑎2 𝑥𝑖
2
+ ⋯ + 𝑎𝑗 𝑥𝑖
𝑗
))2
where: n - # of data points given, 𝑖- the current data point being summed, 𝑗- is
the polynomial order
• The error can be rewritten as:
𝑒𝑟𝑟 =
𝑖=1
𝑛
𝑦𝑖 − 𝑎0 +
𝑖=1
𝑗
𝑎 𝑘 𝑥 𝑘
• find the best line = minimize the error (squared distance) between line and data
points.
13. • Overfit
• over-doing the requirement for the fit to ‘match’ the data trend (order too high).
• Picking an order too high will overfit the data.
• Underfit
• If the order is too low to capture obvious trends in the data