Outline of the lecture:
Introduction of Regression
Application of Regression
Regression Techniques
Types of Regression
Goodness of fit
MATLAB/MATHEMATICA implementation with some example
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.
By DR. SUMMIYA PARVEEN
A short presentation on the topic Numerical Integration for Civil Engineering students.
This presentation consist of small introduction about Simpson's Rule, Trapezoidal Rule, Gaussian Quadrature and some basic Civil Engineering problems based of above methods of Numerical Integration.
This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
A short presentation on the topic Numerical Integration for Civil Engineering students.
This presentation consist of small introduction about Simpson's Rule, Trapezoidal Rule, Gaussian Quadrature and some basic Civil Engineering problems based of above methods of Numerical Integration.
This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
In Numerical analysis, interpolation is a manner of calculating the unknown values of a function for any conferred value of argument within the limit of the arguments. It provides basically a concept of estimating unknown data with the aid of relating acquainted data. The main goal of this research is to constitute a central difference interpolation method which is derived from the combination of Gauss’s third formula, Gauss’s Backward formula and Gauss’s forward formula. We have also demonstrated the graphical presentations as well as comparison through all the existing interpolation formulas with our propound method of central difference interpolation. By the comparison and graphical presentation, the new method gives the best result with the lowest error from another existing interpolationformula.
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
In Numerical analysis, interpolation is a manner of calculating the unknown values of a function for any conferred value of argument within the limit of the arguments. It provides basically a concept of estimating unknown data with the aid of relating acquainted data. The main goal of this research is to constitute a central difference interpolation method which is derived from the combination of Gauss’s third formula, Gauss’s Backward formula and Gauss’s forward formula. We have also demonstrated the graphical presentations as well as comparison through all the existing interpolation formulas with our propound method of central difference interpolation. By the comparison and graphical presentation, the new method gives the best result with the lowest error from another existing interpolationformula.
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
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
Final project report on grocery store management system..pdfKamal Acharya
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.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
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
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
Online aptitude test management system project report.pdf
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