The document summarizes a lecture on regression analysis and curve fitting in mathematical modelling. It introduces regression analysis and its applications. It describes different regression techniques like linear, nonlinear, polynomial and multiple regression. It provides examples of fitting linear and polynomial curves to data using MATLAB. It discusses assessing the goodness of fit using metrics like residual norm and coefficient of determination.
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.
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.
Least Square Optimization and Sparse-Linear SolverJi-yong Kwon
Short slide that explains about the least square problem and its practical solution, including Poisson Image editing example and brief introduction of sparse linear solver.
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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Comparing the methods of Estimation of Three-Parameter Weibull distributionIOSRJM
Weibull distribution has many applications in engineering and plays an important role in reliability. Estimation of the location, scale and shape parameters of this distribution for both censored and non censored samples were considered by several authors. In this paper we compare Graphical oriented methods, “trial and error” approach, the approach of Jiang/Murthy and Maximum likelihood method developed by Bain & Engelhard for sample sets containing uncensored and censored sample. Importance of each method is discussed.
simple linear regression - brief introductionedinyoka
Goal of regression analysis: quantitative description and
prediction of the interdependence between two or more variables.
• Definition of the correlation
• The specification of a simple linear regression model
• Least squares estimators: construction and properties
• Verification of statistical significance of regression model
Least Square Optimization and Sparse-Linear SolverJi-yong Kwon
Short slide that explains about the least square problem and its practical solution, including Poisson Image editing example and brief introduction of sparse linear solver.
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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Castbox: https://castbox.fm/networks/505?country=in
Comparing the methods of Estimation of Three-Parameter Weibull distributionIOSRJM
Weibull distribution has many applications in engineering and plays an important role in reliability. Estimation of the location, scale and shape parameters of this distribution for both censored and non censored samples were considered by several authors. In this paper we compare Graphical oriented methods, “trial and error” approach, the approach of Jiang/Murthy and Maximum likelihood method developed by Bain & Engelhard for sample sets containing uncensored and censored sample. Importance of each method is discussed.
simple linear regression - brief introductionedinyoka
Goal of regression analysis: quantitative description and
prediction of the interdependence between two or more variables.
• Definition of the correlation
• The specification of a simple linear regression model
• Least squares estimators: construction and properties
• Verification of statistical significance of regression model
To get a copy of the slides for free Email me at: japhethmuthama@gmail.com
You can also support my PhD studies by donating a 1 dollar to my PayPal.
PayPal ID is japhethmuthama@gmail.com
To get a copy of the slides for free Email me at: japhethmuthama@gmail.com
You can also support my PhD studies by donating a 1 dollar to my PayPal.
PayPal ID is japhethmuthama@gmail.com
An Approach to Detecting Writing Styles Based on Clustering Techniquesambekarshweta25
An Approach to Detecting Writing Styles Based on Clustering Techniques
Authors:
-Devkinandan Jagtap
-Shweta Ambekar
-Harshit Singh
-Nakul Sharma (Assistant Professor)
Institution:
VIIT Pune, India
Abstract:
This paper proposes a system to differentiate between human-generated and AI-generated texts using stylometric analysis. The system analyzes text files and classifies writing styles by employing various clustering algorithms, such as k-means, k-means++, hierarchical, and DBSCAN. The effectiveness of these algorithms is measured using silhouette scores. The system successfully identifies distinct writing styles within documents, demonstrating its potential for plagiarism detection.
Introduction:
Stylometry, the study of linguistic and structural features in texts, is used for tasks like plagiarism detection, genre separation, and author verification. This paper leverages stylometric analysis to identify different writing styles and improve plagiarism detection methods.
Methodology:
The system includes data collection, preprocessing, feature extraction, dimensional reduction, machine learning models for clustering, and performance comparison using silhouette scores. Feature extraction focuses on lexical features, vocabulary richness, and readability scores. The study uses a small dataset of texts from various authors and employs algorithms like k-means, k-means++, hierarchical clustering, and DBSCAN for clustering.
Results:
Experiments show that the system effectively identifies writing styles, with silhouette scores indicating reasonable to strong clustering when k=2. As the number of clusters increases, the silhouette scores decrease, indicating a drop in accuracy. K-means and k-means++ perform similarly, while hierarchical clustering is less optimized.
Conclusion and Future Work:
The system works well for distinguishing writing styles with two clusters but becomes less accurate as the number of clusters increases. Future research could focus on adding more parameters and optimizing the methodology to improve accuracy with higher cluster values. This system can enhance existing plagiarism detection tools, especially in academic settings.
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In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
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This document will discuss each of the underlying technologies to create and implement an e- commerce website.
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
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NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
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Data Approximation in Mathematical Modelling Regression Analysis and curve fitting
1. Data Approximation in Mathematical
Modelling: Regression Analysis and
Curve Fitting
DR. SUMMIYA PARVEEN
Department of Mathematics
COLLEGE OF ENGINEERING ROORKE (COER)
ROORKEE
summiyaparveen82@gmail.com
Dr. Summiya Parveen 1
2. Outline of the lecture:
Introduction of Regression
Application of Regression
Regression Techniques
Types of Regression
Goodness of fit
MATLAB/MATHEMATICA
implementation with some example
Dr. Summiya Parveen 2
3. Regression
Regression analysis is a form of predictive modelling technique which investigates the
relationship between a dependent (target) and independent variable (s) (predictor). This
technique is used for forecasting, time series modelling and finding the casual effect
relationship between the variables.
Regression analysis is an important tool for modelling and analysing data. Here, we fit a
curve / line to the data points in such a manner that the differences between the distances of
data points from the curve or line is minimized.
Independent variable (x)
Dependentvariable(y)
Year Research &
development
Investment
(millions)
Annual Profit
(millions)
2011 2 20
2012 3 25
2013 5 34
2014 4 30
2015 11 40
2016 5 31
2017 6 25
2019 15 ?
2020 ? 50
Dr. Summiya Parveen 3
4. Applications of Regression Analysis
Agricultural Science
Industrial Production
Environment Science
Business
Health Care
Dr. Summiya Parveen 4
5. Regression Techniques
There are various types of regression techniques available
to make predictions. These techniques are mostly driven
by three metrics (number of independent variables, type of
dependent variables and shape of regression line).
Dr. Summiya Parveen 5
6. Commonly used Types of Regression
Linear Regression
Non - Linear Regression
Polynomial Regression
Multiple Regression
Dr. Summiya Parveen 6
7. Linear Regression
The output of a simple
regression is the coefficient a1
and the constant a0. The
equation is then:
y = a0 + a1 x + e
where
e is the residual error.
a1 is the per unit change in the
dependent variable for each unit
change in the independent
variable. Mathematically:
Dr. Summiya Parveen 7
8. Non-linear Regression
Non-linear functions can also be fitted as regressions.
For examples Power function , Logarithmic function and
Exponential functions.
Dr. Summiya Parveen 8
9. Polynomial Regression
Polynomial equation in m degree
may be taken as :
y = a0 + a1x + a2x2 +....amxm+ e
Here a0 , a1, ……. am
are constant and
e = residual error
Dr. Summiya Parveen 9
10. Multiple Linear Regression
A useful extension of linear regression is the case where
dependent variable y is a linear function of two or more
independent variables
e.g
y = ao + a1x1 + a2x2
We follow the same procedure
y = ao + a1x1 + a2x2 + e
where
e= residual error .
Dr. Summiya Parveen 10
11. Linear Regression
Independent variable (x)
Dependentvariable(y)
The output of a regression is a function that predicts the
dependent variable based upon values of the independent
variable.
Linear regression fits a straight line to the data.
y = a0 + a1 x + e
a0 (y intercept)
a1 = slope
= ∆y/ ∆x
e
Dr. Summiya Parveen 11
12. 12
Fitting a straight line to a
set of paired observations:
(x1, y1), (x2, y2),…,(xn, yn)
yi = a0 + a1 xi + ei
ei = yi - a0 - a1 xi
Here
yi : measured value
ei : error
a1 : slope
a0 : intercept
Linear Regression
e Error
Line equation
y = a0 + a1 x
Dr. Summiya Parveen
13. Best strategy is to minimize the sum of the squares of the residual errors
between the measured-y and the y calculated with the linear model:
Here we need to compute a0 and a1 such that Sr is minimized.
n
i
iir
n
i
modelimeasuredi
n
i
ir
xaayS
yy
eS
1
2
10
1
2
,,
1
2
)(
)(
e Error
Dr. Summiya Parveen 13
14. Least-Square Fit of a Straight Line
00)(2
00)(2
2
101
1
101
iiiiiioi
r
iiioi
o
r
xaxaxyxxaay
a
S
xaayxaay
a
S
Normal equations which can
be solved simultaneously
iiii
ii
xyaxax
yaxna
naa
1
2
0
10
00
(2)
(1)
Since
n
i
ii
n
i
ir xaayeS
1
2
10
1
2
)(:errorMinimize
Dr. Summiya Parveen 14
16. To understand how well the X predicts the Y, we evaluate
Variability in the Y
variable
SSR –> Regression
Variability that is
explained by the
relationship b/w X & Y
+
SSE –> Unexplained
Variability, due to
factors then the
regression
-------------------------------
SST –> Total variability
about the mean
Correlation
Coefficient
r – Strength of the
Relationship
between Y and X
variables
Standard
Error
St Deviation of
error around
the Regression
Line
Residual
Analysis
Validation of
Model
Coefficient of Determination
R Sq - Proportion of explained
variation
Test for Linearity
Significance of the
Regression Model
i.e. Linear Regression
Model
“Goodness” of fit
Dr. Summiya Parveen 16
18. The Coefficient of Determination
The coefficient of determination (R ) is the proportion of the
variability in Y that is explained by the regression equation.
The value of R can range between 0 and 1, and the higher its
value the more accurate the regression model is. It is often
referred to as a percentage.
2
2
Dr. Summiya Parveen 18
20. Standard Error of Regression
The Standard Error of a regression is a measure of its
variability. It can be used in a similar manner to standard
deviation, allowing for prediction intervals.
Standard Error is calculated by taking the square root of the
average prediction error.
Standard Error/Deviation =
where n is the number of observations in the sample and k is
the total number of variables in the model.
If Standard error is low then less number are away from the
mean and if Standard error is high then more number are
away from the mean.
SSE
n - k√
Dr. Summiya Parveen 20
21. Least Squares Fit of a Straight Line:
Example
Fitting a straight line y = a0 + a1 x to the x and y
values given in the following table:
5.119 ii yx
,28 ix 0.24 iy
,1402
ix
4285.3
7
24
4
7
28
yx
428571.3
7
24
4
7
28
yx
xi yi xiyi xi
2
1 0.5 0.5 1
2 2.5 5 4
3 2 6 9
4 4 16 16
5 3.5 17.5 25
6 6 36 36
7 5.5 38.5 49
28 24 119.5 140
Dr. Summiya Parveen 21
22. 1 22
2
0 1
( )
7 119.5 28 24
0.8392857
7 140 28
3.428571 0.8392857 4 0.07142857
i i i i
i i
n x y x y
a
n x x
a y a x
y* = 0.07142857 + 0.8392857 x
Dr. Summiya Parveen 22
23. Error Analysis
9911.2
2
ir eS
932.0868.02
Rr
xi yi
1 0.5
2 2.5
3 2.0
4 4.0
5 3.5
6 6.0
7 5.5
8.5765 0.1687
0.8622 0.5625
2.0408 0.3473
0.3265 0.3265
0.0051 0.5896
6.6122 0.7972
4.2908 0.1993
222
*)( yye)y(y iii
28 24.0 22.7143 2.9911
868.02
t
rt
S
SS
R
7143.22
2
yyS it
Dr. Summiya Parveen 23
24. 9457.1
17
7143.22
1
n
S
s t
y
7735.0
27
9911.2
2
/
n
S
s r
xy
yxy SS /
•The standard deviation (quantifies the spread around the mean):
•The standard error of estimate (quantifies the spread around the
regression line)
Because the linear regression model has good fitness.
Dr. Summiya Parveen 24
26. Required Toolboxes :
A. Curve Fitting Toolbox
B. Statistics Toolbox
C. Spline Toolbox
Dr. Summiya Parveen 26
27. Curve Fitting using inbuilt functions
polyfit(x,y,n)
finds the coefficients of a polynomial P(x) of degree n that fits
the data
It uses least-square minimization
n = 1 (linear fit)
[P] = polyfit(X,Y,N)
returns P, a matrix containing the slope and the x intercept for a
linear fit
[Y] = polyval(P,X)
calculates the Y values for every X point on the line of best fit
Dr. Summiya Parveen 27
28. Curve Fitting Example
• 2nd Order Polynomial Fit:
%read data
[var1, var2] = textread(‘week8_testdata2.txt','%f%f','headerlines',1)
% Calculate 2nd order polynomial fit
P2 = polyfit(var1,var2,2);
Y2 = polyval(P2,var1);
%Plot fit
close all
figure(1)
hold on
plot(var1,var2,'ro')
[sortedvar1, sortind] = sort(var1)
plot(sortedvar1,Y2(sortind),'b*-')Dr. Summiya Parveen 28
30. Curve Fitting Example
• Add 3rd Order Polynomial Fit:
% Calculate 3rd order polynomial fit
P3 = polyfit(var1,var2,3);
Y3 = polyval(P3,var1);
%Add fit to figure
figure(1)
plot(sortedvar1,Y3(sortind),’g*-')
Dr. Summiya Parveen 30
31. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
-3
-2
-1
0
1
2
3
2nd Order Polynomial Fit:
3rd Order Polynomial Fit:
Dr. Summiya Parveen 31
32. Curve Fitting Example
• Add 4th Order Polynomial Fit:
% Calculate 4th order polynomial fit
P4 = polyfit(var1,var2,4);
Y4 = polyval(P4,var1);
%Add fit to figure
figure(1)
plot(sortedvar1,Y4(sortind),’k*-')
Dr. Summiya Parveen 32
33. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
-3
-2
-1
0
1
2
3
2nd Order Polynomial Fit:
3rd Order Polynomial Fit:
4th Order Polynomial Fit:
Dr. Summiya Parveen 33
34. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
-3
-2
-1
0
1
2
3
Assessing Goodness of Fit
Example Solution
% recall var1 contains x values and var2 contains y values of data points
ypred = polyval(P2,var1);
dev = var2 - mean(2);
SST = sum(dev.^2);
resid = var2 - ypred;
SSE = sum(resid.^2);
normr = sqrt(SSE); % residual norm
Rsq = 1 - SSE/SST; % R2 Error
Normr = 5.7436
Rsq = 0.8533
• The residual norm and R2 error indicate goodness of fit
2nd Order Polynomial Fit:
Dr. Summiya Parveen 34
35. Limitations of Polyfit
• Only finds a least squares best polynomial
function fit
• Cannot be used to interpolate curves or fit other
standard functions
• Requires several lines of code and the polyval()
function
Dr. Summiya Parveen 35