This document provides an introduction and overview of linear regression. It begins by explaining what linear regression is using an example of drawing a best fit line through scatter plot data points. It then walks through creating a mock dataset and visualizing it. Next, it demonstrates drawing an initial best fit line by manually choosing slope and bias values. It introduces the mean squared error metric for evaluating fit and uses it to iteratively improve the line. Finally, it derives the gradient descent algorithm for automatically finding the optimal slope and bias values that minimize mean squared error, without manual adjustment of the parameters. Key steps include calculating partial derivatives of the loss function with respect to slope and bias, then updating the parameters in each iteration based on these gradients.
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- The document provides information about statisticshomeworkhelper.com, a service that offers probability and statistics assignment help. It lists their website, email, and phone number for contacting them.
- It then provides an example of a multi-part statistics problem involving hypothesis testing on coin flips and dice data. It asks the reader to conduct various statistical tests and interpret the results.
- Finally, it lists some additional practice problems involving chi-square tests, ANOVA, and other statistical analyses for the reader to work through.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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The Day You Finally Use Algebra: A 3D Math PrimerJanie Clayton
This document provides an overview of various math and programming concepts used for graphics. It begins with an introduction to linear algebra and how it allows performing actions on multiple values simultaneously through matrices. It then discusses trigonometry and how triangles are used as a foundation for 3D graphics. Finally, it shares code for a fragment shader that simulates refraction through a sphere to demonstrate these concepts in action.
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I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
- The document provides information about statisticshomeworkhelper.com, a service that offers probability and statistics assignment help. It lists their website, email, and phone number for contacting them.
- It then provides an example of a multi-part statistics problem involving hypothesis testing on coin flips and dice data. It asks the reader to conduct various statistical tests and interpret the results.
- Finally, it lists some additional practice problems involving chi-square tests, ANOVA, and other statistical analyses for the reader to work through.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I am Joshua M. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from, Michigan State University, USA. I have been helping students with their assignments for the past 6 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
The Day You Finally Use Algebra: A 3D Math PrimerJanie Clayton
This document provides an overview of various math and programming concepts used for graphics. It begins with an introduction to linear algebra and how it allows performing actions on multiple values simultaneously through matrices. It then discusses trigonometry and how triangles are used as a foundation for 3D graphics. Finally, it shares code for a fragment shader that simulates refraction through a sphere to demonstrate these concepts in action.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I am Simon M. I am a Data Analysis Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Nottingham Trent University, UK
I have been helping students with their homework for the past 8 years. I solve assignments related to Data Analysis.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
I am Anthony F. I am a Math Exam Helper at liveexamhelper.com. I hold a Masters' Degree in Maths, University of Cambridge, UK. I have been helping students with their exams for the past 9 years. You can hire me to take your exam in Math.
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This document provides examples and explanations for finding the x- and y-intercepts of linear equations and using intercepts to graph lines. It defines x- and y-intercepts, shows how to find them by setting x or y equal to 0, and explains that intercepts represent important values on the graph like when distance or time is 0. Examples demonstrate finding intercepts, generating points, and graphing lines based on the intercepts.
This document provides examples and explanations for finding the x-intercept and y-intercept of linear equations and using those intercepts to graph the line. It discusses interpreting the intercepts in real world contexts like time to complete a race or amount of items that can be purchased. Students are given examples of finding intercepts, graphing lines by plotting the intercept points, and word problems involving linear equations.
1) The document discusses calculus concepts of derivatives and integrals. Derivatives measure the rate of change of a function, while integrals calculate the area under a function.
2) It provides examples of how to calculate derivatives and integrals, including the rules for derivatives of constants, powers, sums, products, and compositions. Integrals are calculated by dividing the area into rectangles.
3) As an application, it shows how to calculate the economic order quantity by setting the derivative of the total cost function equal to zero to find the minimum.
The document discusses classification algorithms in machine learning. It introduces classification problems using the Iris flower dataset as an example, which contains measurements of Iris flowers to classify them into three species. It then discusses two classic classification algorithms - logistic regression and Gaussian discriminant analysis. Logistic regression uses a sigmoid function to generate predictions, while Gaussian discriminant analysis assumes a Gaussian distribution of the data. The document also demonstrates an application of these algorithms to classify handwritten digits.
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
I am Watson A. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Liberty University, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistics.
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You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
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This document provides an overview of the topics covered in the Numerical Methods course CISE-301. It discusses:
- Numerical methods as algorithms used to obtain numerical solutions to mathematical problems when analytical solutions do not exist or are difficult to obtain.
- Specific topics that will be covered, including solution of nonlinear equations, linear equations, curve fitting, interpolation, numerical integration, differentiation, and ordinary and partial differential equations.
- An introduction to Taylor series and how they can be used to approximate functions, along with examples of Maclaurin series expansions.
- How numerical representations of real numbers like floating point can lead to rounding errors, and the concepts of accuracy and precision in numerical calculations.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
The document defines key terms related to functions including univariate and bivariate data, independent and dependent variables, domain and range, and linear, exponential, quadratic, and step functions. It provides examples of evaluating various functions and finding linear and quadratic models to describe relationships between variables from sets of data points. The overall content describes different types of mathematical functions and how to analyze and model real-world data using functions.
This document discusses Joseph-Louis Lagrange and interpolation. It provides:
1) A brief biography of Joseph-Louis Lagrange, an Italian mathematician who made significant contributions to calculus and probability.
2) A definition of interpolation as producing a function that matches given data points exactly and can be used to approximate values between points.
3) An explanation of Lagrange's interpolation formula for finding a polynomial that fits a set of data points, including an example of applying the formula.
This document provides instructions for a MATLAB assignment with two parts. Part I involves constructing Lagrange interpolants for a given function. Students are asked to create MATLAB function files for Lagrange interpolation and for defining a test function, as well as a script file to test the interpolation. Part II involves solving a system of linear ordinary differential equations and constructing the solution at discrete time points. Students are asked to create a function file to solve the ODE using eigenvalues and eigenvectors, and a script file to test it on a sample problem. Detailed hints are provided for both parts.
Computer Graphics in Java and Scala - Part 1Philip Schwarz
Computer Graphics in Java and Scala - Part 1.
Continuous (Logical) and Discrete (Device) Coordinates,
with a simple yet pleasing example involving concentric triangles.
Scala code: https://github.com/philipschwarz/computer-graphics-50-triangles-scala
Errata:
1. Scala classes TrianglesPanel and Triangles need not be classes, they could just be objects.
I am Stacy W. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Probability.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Probability Assignments.
I am Irene M. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from California, USA. I have been helping students with their homework for the past 6 years. I solve assignments related to Stochastic Processes.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
I am Ronald G. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I have done Ph.D Statistics from New York University, USA. I have been helping students with their statistics assignments for the past 5 years. You can hire me for any of your statistics assignments.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with statistics.
This document provides examples and explanations for finding the x- and y-intercepts of linear equations and using intercepts to graph lines. It defines x- and y-intercepts, shows how to find them by setting x or y equal to 0, and explains that intercepts represent important values on the graph like when distance or time is 0. Examples demonstrate finding intercepts, generating points, and graphing lines based on the intercepts.
This document provides examples and explanations for finding the x-intercept and y-intercept of linear equations and using those intercepts to graph the line. It discusses interpreting the intercepts in real world contexts like time to complete a race or amount of items that can be purchased. Students are given examples of finding intercepts, graphing lines by plotting the intercept points, and word problems involving linear equations.
1) The document discusses calculus concepts of derivatives and integrals. Derivatives measure the rate of change of a function, while integrals calculate the area under a function.
2) It provides examples of how to calculate derivatives and integrals, including the rules for derivatives of constants, powers, sums, products, and compositions. Integrals are calculated by dividing the area into rectangles.
3) As an application, it shows how to calculate the economic order quantity by setting the derivative of the total cost function equal to zero to find the minimum.
The document discusses classification algorithms in machine learning. It introduces classification problems using the Iris flower dataset as an example, which contains measurements of Iris flowers to classify them into three species. It then discusses two classic classification algorithms - logistic regression and Gaussian discriminant analysis. Logistic regression uses a sigmoid function to generate predictions, while Gaussian discriminant analysis assumes a Gaussian distribution of the data. The document also demonstrates an application of these algorithms to classify handwritten digits.
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
I am Watson A. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Liberty University, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
If you are looking for business statistics homework help, Statisticshelpdesk is your rightest destination. Our experts are capable of solving all grades of business statistics homework with best 100% accuracy and originality. We charge reasonable.
This document provides an overview of the topics covered in the Numerical Methods course CISE-301. It discusses:
- Numerical methods as algorithms used to obtain numerical solutions to mathematical problems when analytical solutions do not exist or are difficult to obtain.
- Specific topics that will be covered, including solution of nonlinear equations, linear equations, curve fitting, interpolation, numerical integration, differentiation, and ordinary and partial differential equations.
- An introduction to Taylor series and how they can be used to approximate functions, along with examples of Maclaurin series expansions.
- How numerical representations of real numbers like floating point can lead to rounding errors, and the concepts of accuracy and precision in numerical calculations.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
The document defines key terms related to functions including univariate and bivariate data, independent and dependent variables, domain and range, and linear, exponential, quadratic, and step functions. It provides examples of evaluating various functions and finding linear and quadratic models to describe relationships between variables from sets of data points. The overall content describes different types of mathematical functions and how to analyze and model real-world data using functions.
This document discusses Joseph-Louis Lagrange and interpolation. It provides:
1) A brief biography of Joseph-Louis Lagrange, an Italian mathematician who made significant contributions to calculus and probability.
2) A definition of interpolation as producing a function that matches given data points exactly and can be used to approximate values between points.
3) An explanation of Lagrange's interpolation formula for finding a polynomial that fits a set of data points, including an example of applying the formula.
This document provides instructions for a MATLAB assignment with two parts. Part I involves constructing Lagrange interpolants for a given function. Students are asked to create MATLAB function files for Lagrange interpolation and for defining a test function, as well as a script file to test the interpolation. Part II involves solving a system of linear ordinary differential equations and constructing the solution at discrete time points. Students are asked to create a function file to solve the ODE using eigenvalues and eigenvectors, and a script file to test it on a sample problem. Detailed hints are provided for both parts.
Computer Graphics in Java and Scala - Part 1Philip Schwarz
Computer Graphics in Java and Scala - Part 1.
Continuous (Logical) and Discrete (Device) Coordinates,
with a simple yet pleasing example involving concentric triangles.
Scala code: https://github.com/philipschwarz/computer-graphics-50-triangles-scala
Errata:
1. Scala classes TrianglesPanel and Triangles need not be classes, they could just be objects.
Fundamentals of Multimedia - Vector Graphics.pdfFatihahIrra
This document provides an introduction to vector graphics, contrasting them with raster/bitmap graphics. It discusses how vector graphics use mathematical relationships between points and paths rather than a grid of pixels. Key aspects covered include how vectors describe coordinates and displacements between points, how basic shapes can be defined through vector equations rather than every pixel coordinate, and how vector drawings can be scaled smoothly versus the "staircase effect" in bitmaps. Examples of simple vector code in SVG format are also provided. The document concludes by outlining some more complex vector techniques to be covered next week, such as Bezier curves.
This document provides an overview of NumPy arrays, including how to create and manipulate vectors (1D arrays) and matrices (2D arrays). It discusses NumPy data types and shapes, and how to index, slice, and perform common operations on arrays like summation, multiplication, and dot products. It also compares the performance of vectorized NumPy operations versus equivalent Python for loops.
Linear regression and logistic regression are two machine learning algorithms that can be implemented in Python. Linear regression is used for predictive analysis to find relationships between variables, while logistic regression is used for classification with binary dependent variables. Support vector machines (SVMs) are another algorithm that finds the optimal hyperplane to separate data points and maximize the margin between the classes. Key terms discussed include cost functions, gradient descent, confusion matrices, and ROC curves. Code examples are provided to demonstrate implementing linear regression, logistic regression, and SVM in Python using scikit-learn.
This document summarizes a student project on building machine learning models to recognize handwritten digits. The project involved collecting handwritten digit data, preprocessing the data, implementing logistic regression with gradient descent, evaluating performance on test data, and achieving over 90% accuracy on the test set. The models have applications in areas like banking, postal services, and document digitization.
This document discusses machine learning algorithms for classification problems, specifically logistic regression. It explains that logistic regression predicts the probability of a binary outcome using a sigmoid function. Unlike linear regression, which is used for continuous outputs, logistic regression is used for classification problems where the output is discrete/categorical. It describes how logistic regression learns model parameters through gradient descent optimization of a likelihood function to minimize error. Regularization techniques can also be used to address overfitting issues that may arise from limited training data.
This document summarizes Andrew Ng's lecture notes on supervised learning and linear regression. It begins with examples of supervised learning problems like predicting housing prices from living area size. It introduces key concepts like training examples, features, hypotheses, and cost functions. It then describes using linear regression to predict prices from area and bedrooms. Gradient descent and stochastic gradient descent are introduced as algorithms to minimize the cost function. Finally, it discusses an alternative approach using the normal equations to explicitly minimize the cost function without iteration.
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This document provides a summary of basic MATLAB commands organized into sections:
1) Basic scalar, vector, and matrix operations including declaring variables, performing arithmetic, accessing elements, and clearing memory.
2) Using character strings to print output and combine text.
3) Common mathematical operations like exponents, logarithms, trigonometric functions, rounding, and converting between numeric and string formats.
The document demonstrates how to perform essential tasks in MATLAB through examples and explanations of core commands. It serves as an introductory tutorial for newcomers to learn MATLAB's basic functionality.
Here are the steps to solve this problem numerically in MATLAB:
1. Define the 2nd order ODE for the pendulum as two first order equations:
y1' = y2
y2' = -sin(y1)
2. Create an M-file function pendulum.m that returns the right hand side:
function dy = pendulum(t,y)
dy = [y(2); -sin(y(1))];
end
3. Use an ODE solver like ode45 to integrate from t=0 to t=6pi with initial conditions y1(0)=pi, y2(0)=0:
[t,y] = ode45
1) Regression models analyze data to find patterns and relationships that can be used to predict future trends or values.
2) A linear regression finds the line of best fit to model the relationship between two variables in a data set.
3) The document demonstrates how to create a linear regression model by plotting sample data, determining the best fit line, calculating the line's slope and y-intercept, and writing the equation in slope-intercept form.
This document provides an introduction to machine learning concepts including supervised and unsupervised learning. It discusses linear regression with one variable and multiple variables. For linear regression with one variable, it describes the hypothesis function, cost function, gradient descent algorithm, and makes predictions using a housing dataset. For multiple variables, it introduces feature normalization and applies the concepts to predict housing prices based on size, bedrooms and price in a real estate dataset. The document provides code examples to implement the algorithms.
This document discusses different types of plots that can be created in MATLAB, including bar charts, contour plots, and 3D surface plots. It provides examples of the code needed to generate each type of plot using built-in MATLAB functions like bar, contour, meshgrid, and surf. For bar charts, the bar function is used to plot scores of 10 students. For contour plots, the contour function plots contour lines for a function of two variables defined on a meshgrid. For 3D surface plots, the surf function plots a 3D surface defined by a function of two variables on a meshgrid.
The document discusses linear regression for predicting salary based on years of experience. It introduces gradient descent for linear regression, which iteratively updates the slope and intercept parameters (θ1 and θ2) to minimize cost and improve predictions. Gradient descent takes steps in the direction of the steepest descent down the cost function landscape. The learning rate determines step sizes and must be optimized for accurate predictions within a reasonable time.
This document provides an introduction and overview of machine learning algorithms. It begins by discussing the importance and growth of machine learning. It then describes the three main types of machine learning algorithms: supervised learning, unsupervised learning, and reinforcement learning. Next, it lists and briefly defines ten commonly used machine learning algorithms including linear regression, logistic regression, decision trees, SVM, Naive Bayes, and KNN. For each algorithm, it provides a simplified example to illustrate how it works along with sample Python and R code.
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docxagnesdcarey33086
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
Joe Bob
Mon lab: 4:30-6:50
Lab 3
Exercise 1
(a) Create function M-file for banded LU factorization
function [L,U] = luband(A,p)
% LUBAND Banded LU factorization
% Adaptation to LUFACT
% Input:
% A diagonally dominant square matrix
% Output:
% L,U unit lower triangular and upper triangular such that LU=A
n = length(A);
L = eye(n); % ones on diagonal
% Gaussian Elimination
for j = 1:n-1
a = min(j+p.
This document provides an overview of linear regression and logistic regression concepts. It begins with an introduction to linear regression, discussing finding the best fit line to training data. It then covers the loss function and gradient descent optimization algorithm used to minimize loss and fit the model parameters. Next, it discusses logistic regression for classification problems, covering the sigmoid function for hypothesis representation and interpreting probabilities. It concludes by discussing feature scaling techniques like normalization and standardization to prepare data for modeling.
This document provides an introduction to MATLAB. It discusses that MATLAB is a high-performance language for technical computing that integrates computation, visualization, and programming. It can be used for tasks like math and computation, algorithm development, modeling, simulation, prototyping, data analysis, and scientific graphics. MATLAB uses arrays as its basic data type and allows matrix and vector problems to be solved more quickly than with other languages. The document then provides examples of entering matrices, using basic MATLAB commands and functions, plotting graphs, and writing MATLAB code in M-files.
Calculus Application Problem #3 Name _________________________.docxhumphrieskalyn
Calculus Application Problem #3 Name __________________________________________
The Deriving Dead! Due at the beginning of class ______________________
Introduction: Imagine that you are one of many people at a “party” and that, unknown to everyone else, one
person was bitten by a zombie on the way to the party! How quickly will the “zombiepocalypse” spread, and
what are the chances that you will leave the party as a zombie? The objective of this activity is to create a
mathematical model that describes the spread of a disease (such as a zombie virus) in a closed environment, and
then apply calculus concepts to this mathematical model.
Collecting the Data:
Let’s collect some data from an activity that will simulate the spread of a communicable disease over a period of
time, divided into “stages”.
The number of people in our “closed environment” is ________________
1
1
131211109876543210
Number
of Total
Infected
Number
of Newly
Infected
Stage
Number
Applying Calculus to the Data:
1. Using the data from the chart, make a scatterplot of the "Stage Number" (in L1) vs. the "Number of Total
Infected" (in L2). Sketch the scatterplot below. Connect the data points to create a continuous function for Y(t).
2. Using the data that was collected in the activity, answer the following questions about the derivative
function Y’(t), which represents the instantaneous rate of change of the number of infected at any stage.
Consider the domain to be [ 0 , 13 ].
a. When, if ever, is Y’(t) positive? ____________________________________
b. When, if ever, is Y’(t) negative? ___________________________________
c. When, if ever, is Y’(t) increasing? ____________________________________
d. When, if ever, is Y’(t) decreasing? ____________________________________
e. From your answers above, sketch a graph of Y’(t) below.
f. The t-value where Y’(t) changes from increasing to decreasing is the inflection point on Y(t).
According to the data in the chart, this occurs when t = _________, and the corresonding “y-value”
is ________.
(Note: We will check this later in the problem!)
Finding a Logistic Function that Models the Data
3. Since the data (should) appear to be a model for a logistic function, we need to find a function in the
form:
Y(t ) =
c
1 + a ⋅ e− b⋅t
,
where t represents the stage number and Y(t) represents the total number of infected people in stage t.
Therefore, we need to find values for the three constants a, b, and c. The value of c should be easy. For our
activity,
c = _________
To find a, use the initial point ( 0 , 1 ). Substitute this ordered pair, with the value of c into our logistic model and
solve for a. Show your work below.
a = _______________
To find b, the last constant in the model, we need another ordered pair. Let’s use an ordered pair near the middle
of the data, say during Stage #7.
Record this ordered pair: ( 7 , _________)
Substitut ...
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
1. Chapter 1: Linear Regression
Coder’s Guide to Deep Learning
Akmel Syed
2. Linear Regression
For our very first lesson, we'll go over something called linear regression. What is it exactly? Linear regression is
just a fancy term for something which you probably somewhat already covered in highschool. Do you remember
best fit lines in highschool math? It was you trying to estimate and draw a straight line through a bunch of
points. These points were plotted on a graph and were arranged in a way which appeard that they were going in
the same direction. I hope you're going "oh yeah, I remember that!" Because that's linear regression! Let's go
through it with some code...
1.1 Making a Mock Dataset
Before we start, let's import a couple libraries which we'll use to create our mock dataset. Sklearn (short for
scikit-learn) is a very widely used machine learning library in Python. We won't be covering it in this book, but it
has a nice built-in function (specifically it's "make_regression" function) for easily making mock datasets. As for
matplotlib, it's an easy to use barebones data science library in Python to plot graphs.
In [1]:
from sklearn.datasets import make_regression
import matplotlib.pyplot as plt
Now that we've imported the libraries, let's use sklearn's make_regression function to make our mock dataset.
Don't worry too much about the details regarding the arguments being passed to the function. All you need to
know is that we're going to make 100 mock data points. Each point has a distinct x and y coordinate.
In [2]:
X, Y = make_regression(n_samples=100, n_features=1, bias=5, noise=5, random_state=762)
Y = Y.reshape(-1, 1)
Now that we have our mock data, let's use matplotlib to visualize it.
In [3]:
plt.scatter(X, Y)
plt.show()
I'm hoping that what you're seeing in the plot above is somewhat what you expected to see. What you're seeing
is 100 distinct points, which all seem to be going in the same direction. If you squint your eyes, you'll see that the
points on the graph somewhat make a shape which looks kind of like a straight line.
1.2 Drawing Our Best Fit Line
Now, it's almost time to draw our best fit line. Before we can do that, let's just quickly go over the definition of a
3. Now, it's almost time to draw our best fit line. Before we can do that, let's just quickly go over the definition of a
line in math. If you remember from highschool math, a line can be define as y=m*x+b, where m represents the
slope, x represents a position on the x-axis and b represents the bias. If none of that made any sense, that's ok.
Playing with the code below should clear it up. I've started you off with values for m and b, but to really
understand how m and b effect the red line below, make sure you give them different values and re-run the code.
Change the values for m and b individually so that you can see how they effect the line.
In [4]:
m, b = 13, 4
y_pred = []
for x in X:
y_pred.append(m*float(x) + b)
plt.scatter(X, Y)
plt.plot(X, y_pred, 'r')
plt.show()
You'll see in the code above that we're plotting 2 different things - the actual points and the line. The line is
defined by our formula y=m*x+b. We loop over each x value in order to get our y value of the line at that point.
There your have it, linear regression!... Well, kinda...
1.3 Finding the Best Fit
So we kinda sorta found a best fit line above. It was us just guessing random values for m and b which made it
look like the line is going through the points on the graph in a balanced way - in a way which we think represents
the general shape of our data well enough that we can take it to be our "best fit" line.
Now, how would you code it so that all of this happens automatically and your algorithm finds a good best fit
line for you? The way I've done it below is semi-automated. I start with the slope (m) and the bias (b) set to 0 and
then I keep incrementing them during every iteration (epoch) by values, which I randomly chose. I decided to set
my last iteration (epoch) to be at a number, not too high and not too low, which I think gives me a good best fit
line. This time, don't play with the values for the slope and the bias, instead, play with the number of epochs to
see how it effects the output.
Just on the side, I'm importing 2 more libraries which are used for demo purposes only. These libraries (time
and IPython) have nothing to do with the actual algorithm. I placed the functions from these 2 libraries between
comments in the actual algorithm, so that you can ignore that part.
In [5]:
import time
from IPython import display
In [6]:
slope = 0
bias = 0
epochs = 15
for epoch in range(epochs):
4. y_pred = []
for x in X:
y_pred.append(slope*float(x) + bias)
######demo purpose only#####
display.display(plt.gcf())
display.clear_output(wait=True)
##########plotting##########
plt.scatter(X, Y)
plt.plot(X, y_pred, 'r')
plt.title('slope = {0:.1f}, bias = {1:.1f}'.format(slope, bias))
plt.show()
###################
slope += 1
bias += 0.3
1.4 Mean Squared Error
So far, we've come to realize a couple things. We saw that the slope and the bias are the numbers which
influence the line and that linear regression is synonymous to a best fit line. The question I want you to think
about now is, what is it that you're looking at to determine what is a best fit line? In your mind, what are you
doing? How I'd define it is that you're doing a comparison of the "best fit" line and the points on the graph. The
further away the line is to the points, the less likely that line is to be considered a "best fit" line. Let's look at an
example. Run the 2 cells below to output the graphs used for this example.
In [7]:
m, b = 14, 4
y_pred = []
for x in X:
y_pred.append(m*float(x) + b)
plt.scatter(X, Y)
plt.plot(X, y_pred, 'r')
plt.title('Figure 1.4.1')
plt.show()
5. In [8]:
m, b = 14, -15
y_pred = []
for x in X:
y_pred.append(m*float(x) + b)
plt.scatter(X, Y)
plt.plot(X, y_pred, 'r')
plt.title('Figure 1.4.2')
plt.show()
Looking at Figure 1.4.1 and Figure 1.4.2, which would you deem to be a better fit? Figure 1.4.1 is clearly the
better fit. Why? Because, in general, it's closer to the points.
Is there a mathematical way to get one number to see how far, on average, the line is to the points? Yes, there is!
We're going to use something known as a loss function. That loss function we'll being using for linear regression
is the mean squared error (MSE). For each point, it finds the distance with its respective x value on the line. The
equation does that for all the points and then gets the average.
Let's write it out in a function.
In [9]:
def mse(y, y_pred): ## mean squared error
summation = 0. ## variable for holding the sum of the distanc
es
n = y.shape[0] ## get the amount of points
for i in range(n): ## for each point
summation += (y[i]-y_pred[i])**2 ## get the distance between the point and it's
respective x value
## on the "best fit" line and square it
return float(summation/n) ## return the average distance aas 1 number
The above seems all good and well, but a visualization would probably exaplain it better. Run the cell below to
see our MSE in action. The graph on the left is the graph with our regression points and the graph on the right is
the MSE value at each epoch.
In [10]:
epoch_loss = []
slope = 0
bias = 0
for epoch in range(25):
y_pred = []
for x in X:
y_pred.append(slope*float(x) + bias)
loss = mse(Y, y_pred)
6. epoch_loss.append(loss)
######demo purpose only#####
display.display(plt.gcf())
display.clear_output(wait=True)
##########plotting##########
fig, (ax0, ax1) = plt.subplots(ncols=2, constrained_layout=True)
ax0.scatter(X, Y)
ax0.plot(X, y_pred, 'r')
ax0.set_title('slope = {0:.1f}, bias = {1:.1f}'.format(slope, bias))
ax1.set_title('mse = {0:.2f}'.format(loss))
ax1.plot(epoch_loss)
plt.show()
time.sleep(1)
###################
slope += 1
bias += 0.3
Now that we have a good understanding of our loss function, we see how much value it can bring us in fitting
our best fit line with confidence. We see that the lowest value for MSE gives us a really good best fit line. The
question now is, how do we automate it so that our algorithm finds the optimal MSE without us having to
manually increment the slope and bias values?
Before we move on, just a reminder, us incrementing the slope by 1 and bias by 0.3 at each epoch was manually
chosen. Those values are fine for this mock dataset, but most probably will not hold true for a different dataset.
Those incrementation values are what we're trying to have our algorithm automatically find.
1.5 Gradient Descent
We've finally made it to gradient descent! This is the part we've been waiting for. It allows us to automatically
find the correct values to update the slope and bias. If you were looking at the U-shaped MSE graph in 1.4 and
thinking calculus, then you're one step ahead. Now, I'll be honest, my calculus isn't the best, so even I was a
little scared by this, but I promise, just stick with it and when we get to PyTorch, you won't have to deal with
calculus again.
The good part is that the calculus for linear regression is very straightforward, but before we get to it, let's just
recap the steps we've taken to get to this point up until now:
1. We made our mock dataset and plotted it.
2. We randomly chose our initial values for our slope and bias for our best fit line.
3. We plotted a best fit line with our values for the slope and bias.
4. At every epoch, we evaluated our line with respect to our MSE.
5. We assigned random values to our slope and bias to update by. This step is what we intend to use calculus
on in order to be able to automate it and remove the random part.
6. We updated our slope and bias, in order to improve our MSE.
<Figure size 432x288 with 0 Axes>
7. 7. Repeat steps 3-6 until satisfied with the MSE.
If you're still wondering how calculus will help us in our quest to automatically find the lowest MSE, then let's
just take a moment to clarify it. If you remember our MSE graph in 1.4, then you'll remember that the graph
looked like a valley - the optimal value being at the bottom of the valley. Calculus is just a tool to calculate the
slope at a given point. Understanding the steepness of the slope and the direction of it allows us to understand
in what direction we should be heading to be able to get to the bottom of the valley. Us slowly "descending" the
valley is why this process is known as gradient descent.
In this book, we won't cover the rules of calculus, but I do suggest that you go over it, if you haven't alread.
There are many good resources online. Regardless, for the purposes of our exercise, I'll be providing you with
the formulae which would be derived using partial derivates (i.e. calculus).
Starting off, our equation for our MSE is (y-y_pred)^2, where y represents the y-values of our mock dataset and
y_pred represents the y-values for our "best fit" line. The derivative for that ends up to be (y-y_pred).
Now, we get to the equation of our line. We said it is y = (slope * X) + bias. The derivate for the slope ends up
being X and the derivative for the bias ends up being 1.
Putting the derivatives for both equations above together, we end up with 2 equations.
1. The derivative of the MSE with respect to the slope is X * (y-y_pred).
2. The derivative of the MSE with respect to the bias is 1 * (y-y_pred), which is the same as just (y-y_pred).
Awesome! We now have all the equations we need to automatically fit this line, and be sure that it is infact a
best fit for our dataset. Now let's run the cell below and see it in action!
In [11]:
epoch_loss = []
slope = 0.
bias = 0.
learning_rate = 1e-3
n = X.shape[0]
for epoch in range(50):
y_pred = []
for x in X:
y_pred.append(slope*float(x) + bias)
loss = mse(Y, y_pred)
epoch_loss.append(loss)
######demo purpose only#####
display.display(plt.gcf())
display.clear_output(wait=True)
##########plotting##########
fig, (ax0, ax1) = plt.subplots(ncols=2, constrained_layout=True)
fig.suptitle('epoch = {0}'.format(epoch))
ax0.scatter(X, Y)
ax0.plot(X, y_pred, 'r')
ax0.set_title('slope = {0:.1f}, bias = {1:.1f}'.format(slope, bias))
ax1.set_title('mse = {0:.2f}'.format(loss))
ax1.plot(epoch_loss)
plt.show()
time.sleep(1)
###################
###derivatives of mse with respect to slope and bias###
###zero-ing out the gradients###
D_mse_wrt_slope = 0.
D_mse_wrt_bias = 0.
###performing back propagtion for each point on the graph###
for i in range(n):
D_mse_wrt_slope += float(X[i]) * (float(Y[i]) - float(y_pred[i]))
D_mse_wrt_bias += float(Y[i]) - float(y_pred[i])
8. ###updating the slope and bias###
slope += learning_rate * D_mse_wrt_slope
bias += learning_rate * D_mse_wrt_bias
Wow, that was cool! We could have probably stopped at 20 epochs, but I wanted to show how robust our
calculations actually are. No matter how many epochs you go for, it just keeps getting better and better. The
power of math. Who knew all those highschool math courses were actually worth something?
You're probably looking at the code and realizing a few things. The first thing is that there's a variable added
known as the learning_rate. What is that? So remeber I told you that calculus allows us to find out the how steep
the slope is at a given point and hence guides us in descending down to the valley? Well, the learning rate is just
a number which tells our algorithm how big the step should be in our desired direction. It let's our algorithm
know if we should be tip toeing down the slope, or if we should be running down.
Try different values for the learning rate and re-run the cell to see how it effects our algorithm. Try the value 1e-2
and then try 1e-4. Also, how did I come up with 1e-3? Unfortunately, I lied and not everything is 100% automated.
That value was chosen via trial and error, but incrementing/decrementing with 10 to the power of a number is a
good rule to go by.
The other thing you're probably seeing is that we set the partial derivate variables (i.e. gradients) to 0 during
every epoch. This is an important step to make sure your gradients are adding up. Try removing that and seeing
how it effects the algorithm
1.6 No Loops!
So far in this chapter, we've used loops to do our summations. We've also used loops to calculate our y values
for our regression line (i.e. best fit line). A little secret about machine learning practitioners is that we hate loops.
We try every trick in the book to get away from loops. Why? Because it's computationally expensive and it's not
parallelizable.
In order to get away from loops, we're going to import a famous library known as NumPy. It's a very powerful
library used for linear algebra.
In [12]:
import numpy as np
Now that we've imported it, let's rewrite our MSE function using NumPy.
In [13]:
def mse(y, y_pred): ##mean squared error
return np.sum((y - y_pred)**2)/y.shape[0]
That's it! Using NumPy's sum function, we were able to reduce our summation loop into only 1 line!
<Figure size 432x288 with 0 Axes>
9. That's it! Using NumPy's sum function, we were able to reduce our summation loop into only 1 line!
Let's see how we can implement the rest of the code using NumPy and remove the loops.
In [14]:
epoch_loss = []
slope = 0.
bias = 0.
learning_rate = 1e-3
for epoch in range(50):
##changed##
y_pred = slope*X + bias
###########
loss = mse(Y, y_pred)
epoch_loss.append(loss)
######demo purpose only#####
display.display(plt.gcf())
display.clear_output(wait=True)
##########plotting##########
fig, (ax0, ax1) = plt.subplots(ncols=2, constrained_layout=True)
fig.suptitle('epoch = {0}'.format(epoch))
ax0.scatter(X, Y)
ax0.plot(X, y_pred, 'r')
ax0.set_title('slope = {0:.1f}, bias = {1:.1f}'.format(slope, bias))
ax1.set_title('mse = {0:.2f}'.format(loss))
ax1.plot(epoch_loss)
plt.show()
time.sleep(1)
###################
###derivatives of mse with respect to slope and bias###
##changed##
D_mse_wrt_slope = np.sum(X * (Y - y_pred))
D_mse_wrt_bias = np.sum(Y - y_pred)
###########
slope += learning_rate * D_mse_wrt_slope
bias += learning_rate * D_mse_wrt_bias
Amazing! Without loops, if we don't count the lines used for plotting, we were able to perform linear regression
using gradient descent in only 10 lines!
There's two things I wish to point out before ending this chapter. The first point is that you probably realized that
I'm not explicity zero-ing out the gradients. The reason for that is because we don't need to. Since we're not
using loops anymore, the new values for the gradients just replace the old ones. The second point is that you'll
see that I've also gotten rid of the loop for our regression line equation. How could I do that? Well, because of
<Figure size 432x288 with 0 Axes>
10. the parallelization of the equation (essentially means that the same equation is happening over and over again in
a loop format), it's an excellent candidate for NumPy. In the background, NumPy is treating it as a matrix
calculation, and if you know about linear algebra, you know that for every row in a matrix, all the calculations can
be done simaltaneously.
1.7 Linear Regression's Computation Graphs
Below, you should find the computation graph for linear regression. It's a nice and simple diagram which sums
up the steps to linear regression using gradient descent. The process starts at the left and goes to the right.
You'll see that the 2 circles on the far left represent our inputs, the circle in the middle is our regression line
equation and the circle on the far right represents our loss function, which is MSE for this case. Going left to
right is known as forward propogation and then going right to left (which is done using calculus) is known as
backwards propogation.
Now that you've had a chance to look at the diagram above, you must be thinking where'd the 1 come from?
We'll address that in Chapter 2: Logisitc Regression.