1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
The first report of Machine Learning Seminar organized by Computational Linguistics Laboratory at Kazan Federal University. See http://cll.niimm.ksu.ru/cms/lang/en_US/main/seminars/mlseminar
Covers supervised learning and discriminative algorithms. Includes: Linear Regression, The LMS Algorithm, Probabalistic interpretations, Classification, Logistic Regression, Underfitting and Overfitting.
The first report of Machine Learning Seminar organized by Computational Linguistics Laboratory at Kazan Federal University. See http://cll.niimm.ksu.ru/cms/lang/en_US/main/seminars/mlseminar
Covers supervised learning and discriminative algorithms. Includes: Linear Regression, The LMS Algorithm, Probabalistic interpretations, Classification, Logistic Regression, Underfitting and Overfitting.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning. Part 1 covers: 1. A brief history of Random Matrix Theory, 2. Classical Random Matrix Ensembles (basic building blocks)
Optimizing a New Nonlinear Reinforcement Scheme with Breeder genetic algorithminfopapers
Florin Stoica, Dana Simian, Optimizing a New Nonlinear Reinforcement Scheme with Breeder genetic algorithm, Proceedings of the Recent Advances in Neural Networks, Fuzzy Systems & Evolutionary Computing,13-15 June 2010, Iasi, Romania, ISSN: 1790-2769, ISBN: 978-960-474-194-6, pp. 273-278
Estimation of Parameters and Missing Responses In Second Order Response Surfa...inventionjournals
This article is an attempt to explore missing responses in second order response design model
using Expected Maximization algorithm with and without imposing restrictions on the design matrix towards or
thogonality are derived and are implemented with suitable examples. The properties of estimated parameters
and estimated responses are also studied and findings are presented in detail at the end of the study.
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 Assignment.
Inference for stochastic differential equations via approximate Bayesian comp...Umberto Picchini
Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
A 3hrs intro lecture to Approximate Bayesian Computation (ABC), given as part of a PhD course at Lund University, February 2016. For sample codes see http://www.maths.lu.se/kurshemsida/phd-course-fms020f-nams002-statistical-inference-for-partially-observed-stochastic-processes/
Opening of our Deep Learning Lunch & Learn series. First session: introduction to Neural Networks, Gradient descent and backpropagation, by Pablo J. Villacorta, with a prologue by Fernando Velasco
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning. Part 1 covers: 1. A brief history of Random Matrix Theory, 2. Classical Random Matrix Ensembles (basic building blocks)
Optimizing a New Nonlinear Reinforcement Scheme with Breeder genetic algorithminfopapers
Florin Stoica, Dana Simian, Optimizing a New Nonlinear Reinforcement Scheme with Breeder genetic algorithm, Proceedings of the Recent Advances in Neural Networks, Fuzzy Systems & Evolutionary Computing,13-15 June 2010, Iasi, Romania, ISSN: 1790-2769, ISBN: 978-960-474-194-6, pp. 273-278
Estimation of Parameters and Missing Responses In Second Order Response Surfa...inventionjournals
This article is an attempt to explore missing responses in second order response design model
using Expected Maximization algorithm with and without imposing restrictions on the design matrix towards or
thogonality are derived and are implemented with suitable examples. The properties of estimated parameters
and estimated responses are also studied and findings are presented in detail at the end of the study.
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 Assignment.
Inference for stochastic differential equations via approximate Bayesian comp...Umberto Picchini
Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
A 3hrs intro lecture to Approximate Bayesian Computation (ABC), given as part of a PhD course at Lund University, February 2016. For sample codes see http://www.maths.lu.se/kurshemsida/phd-course-fms020f-nams002-statistical-inference-for-partially-observed-stochastic-processes/
Opening of our Deep Learning Lunch & Learn series. First session: introduction to Neural Networks, Gradient descent and backpropagation, by Pablo J. Villacorta, with a prologue by Fernando Velasco
A set of notes prepared for an introductory machine learning course, assuming very limited linear algebra background, because all linear algebra operations are fully written out. These notes go into thorough derivations of the generalized linear regression formulation, demonstrating how to write it out in matrix form.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
2. Supervised Learning Problem
Linear Regression / One Feature
1 General
Two definitions of Machine Learning are offered. Arthur Samuel described it as: "the field of study that gives com-
puters the ability to learn without being explicitly programmed." This is an older, informal definition. Tom Mitchell
provides a more modern definition: "A computer program is said to learn from experience E with respect to some
class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experi-
ence E."
In supervised learning, we are given a data set and already know what our correct output should look like, having
the idea that there is a relationship between the input and the output. Supervised learning problems are categorized
into "regression" and "classification" problems. In a regression problem, we are trying to predict results within a con-
tinuous output, meaning that we are trying to map input variables to some continuous function. In a classification
problem, we are instead trying to predict results in a discrete output. In other words, we are trying to map input
variables into discrete categories.
2
3. Supervised Learning Problem
Linear Regression / One Feature
2 Data Set
We define a input set Ω with one bias element x0 and one feature element x1 and a output set Υ with one output
element y1.
Ω = {x0;x1} (2.1)
Υ = {y1} (2.2)
An arbitrary element u ∈ Ω and v ∈ Υ are vectoriezed by m rows.
x0 =
x(1)
0
x(2)
0
x(3)
0
...
x(i)
0
...
x(m)
0
=
1
1
1
...
1
...
1
(2.3)
x1 =
x(1)
1
x(2)
1
x(3)
1
...
x(i)
1
...
x(m)
1
(2.4)
y1 =
y(1)
1
y(2)
1
y(3)
1
...
y(i)
1
...
y(m)
1
(2.5)
We define a set Π with m tuple called training set.
Π = {(1,x(1)
1 , y(1)
1 ); (1,x(2)
1 , y(2)
1 ); (1,x(3)
1 , y(3)
1 ); ... ;(1,x(i)
1 , y(i)
1 ); ... ;(1,x(m)
1 , y(m)
1 )} (2.6)
3
4. Supervised Learning Problem
Linear Regression / One Feature
For a better intuition of the training set we can write it in a table called training table. Every row means one training
example. In total we have m training examples.
1 x(1)
1 y(1)
1
1 x(2)
1 y(2)
1
1 x(3)
1 y(3)
1
...
...
...
1 x(i)
1 y(i)
1
...
...
...
1 x(m)
1 y(m)
1
Table 2.1 Training table
We define the discrete training function t(x(i)
1 ):
t : {x(1)
1 ;x(2)
1 ;x(3)
1 ;... ;x(i)
1 ;... ;x(m)
1 } → {y(1)
1 ; y(2)
1 ; y(3)
1 ;... ; y(i)
1 ;... ; y(m)
1 } (2.7)
x(i)
1 → y(i)
1
We define a matrix T called training matrix. We also define the input matrix X and the output matrix Y.
T =
1 x(1)
1 y(1)
1
1 x(2)
1 y(2)
1
1 x(3)
1 y(3)
1
...
...
...
1 x(i)
1 y(i)
1
...
...
...
1 x(m)
1 y(m)
1
; (2.8)
X =
1 x(1)
1
1 x(2)
1
1 x(3)
1
...
...
1 x(i)
1
...
...
1 x(m)
1
; Y =
y(1)
1
y(2)
1
y(3)
1
...
y(i)
1
...
y(m)
1
(2.9)
4
5. Supervised Learning Problem
Linear Regression / One Feature
In MATLAB we load our data set and initialize variable T, X and Y
data = load(’example.txt’);
m = length(data);
T = [ones(m, 1), data(:,1:2)];
X = [ones(m, 1), data(:,1)];
Y = [data(:,2)];
5
6. Supervised Learning Problem
Linear Regression / One Feature
3 Hypothesis Function
To describe the supervised learning problem - linear regression -, our goal is, given a training set Π, to learn a linear
function. We are defining the continious hypothesis function h(x1):
h : R → R (3.1)
x1 → θ0 +θ1 · x1
so that h(x1) is a good predictor for a arbitrary input variable. For historical reasons the function is called hypothesis.
In other words we are looking the best values for θ0 and θ1. We define the parameter vector θ
θ =
θ0
θ1
(3.2)
and the hypothesis-input vector x
x =
1
x1
(3.3)
So we can write our hypothesis function h(x1) in vectorized form.
h(x1) = θT
· x = θ0 θ1 ·
1
x1
= θ0 +θ1 · x1 (3.4)
In MATLAB we initialize the parameter vector θ and start with a arbitrary value for θ0 and θ1 (preferably equal 0).
theta = zeros(size(X,2),1);
6
7. Supervised Learning Problem
Linear Regression / One Feature
4 Cost Function
We can measure the accuracy of our hypothesis function h(x1) by using a cost function J(θ0,θ1). This takes an average
difference (actually a fancier version of an average) of all the results of the hypothesis dependet on the feature value
x(i)
1 ’s and the output value y(i)
1 ’s. This function is otherwise called the "Squared error function", or "Mean squared
error". The mean is halved 1/2 as a convenience for the computation of the gradient descent, as the derivative term of
the square function will cancel out the 1/2 term.
J(θ0,θ1) =
1
2·m
·
m
i=1
(h(x(i)
1 )− y(i)
1 )2
(4.1)
= ···
=
1
2·m
· m ·θ0
2
+(x1
(1)2
+ x1
(2)2
+···+ x1
(m)2
)·θ1
2
+(2· x1
(1)
+2· x1
(2)
+···+2· x1
(m)
)·θ0 ·θ1
+ (−2· y1
(1)
−2· y1
(2)
−···−2· y1
(m)
)·θ0
+ (−2· x1
(1)
· y1
(1)
−2· x1
(2)
· y1
(2)
−···−2· x1
(m)
· y1
(m)
)·θ1
+ y1
(1)2
+ y1
(2)2
+···+ y1
(m)2
We define the coefficients
a := m (4.2)
b := (x1
(1)2
+ x1
(2)2
+···+ x1
(m)2
) (4.3)
c := (2· x1
(1)
+2· x1
(2)
+···+2· x1
(m)
) (4.4)
d := (−2· y1
(1)
−2· y1
(2)
−···−2· y1
(m)
) (4.5)
e := (−2· x1
(1)
· y1
(1)
−2· x1
(2)
· y1
(2)
−···−2· x1
(m)
· y1
(m)
) (4.6)
f := (y1
(1)2
+ y1
(2)2
+···+ y1
(m)2
) (4.7)
and we can write
J(θ0,θ1) =
1
2·m
·(a ·θ0
2
+b ·θ1
2
+c ·θ0 ·θ1 + d ·θ0 +e ·θ1 + f ) (4.8)
In algebra we call this function a quadratic polynomial with 2 variables. Note, that the cost function for linear regres-
sion is always going to be a bowl-shaped function. Means, that J(θ0,θ1) doesn’t have any local optima except for the
one global optima.
Figure 4.1 Example for a bowl-shaped function
7
8. Supervised Learning Problem
Linear Regression / One Feature
In MATLAB we initialize the cost function J(θ0 = 0,θ1 = 0) with our given input matrix X, given output matrix Y and an
arbitrary parameter vector θ
J = 1/(2*m) * sum( (X*theta - Y).ˆ2 );
8
9. Supervised Learning Problem
Linear Regression / One Feature
5 Gradient Descent Algorithm
So we have our hypothesis function and we have a way of measuring how well it fits into the data. Now we need to
estimate the parameters in the hypothesis function. That’s where gradient descent comes in. Imagine that we graph
our hypothesis function based on its parameter theta zero and theta one. Aactually we are graphing the cost function
as a function of the parameter estimates. We will know that we have succeeded when our cost function is at the very
bottom of the graph. The way we do this is by taking the derivative (the tangential line to a function) of our cost
function.The slope of the tangent is the derivative at that point and it will give us a direction to move towards. We
make steps down the cost function in the direction with the steepest descent. The size of each step is determined by
the parameter α, which is called the learning rate. A smaller α would result in a smaller step and a larger α results
in a larger step. The direction in which the step is taken is determined by the partial derivative of the cost function
∂
∂θj
J(θ0,θ1). Depending on where the parameter vector θ starts on the graph.
With other words, we want an efficient algorithm to find the value of θ0 and θ1. The gradient descent algorithm is:
Keep changing θ0 and θ1 until we end up at the global minimum. Remember: We start at θ0 = 0 and θ1 = 0.
θtemp_0 := θ0 −α·
∂
∂θ0
J(θ0,θ1) = θ0 −α·
1
m
·
m
i=1
(hθ(x(i)
1 )− y(i)
1 ) (5.1)
θtemp_1 := θ1 −α·
∂
∂θ1
J(θ0,θ1) = θ1 −α·
1
m
·
m
i=1
(hθ(x(i)
1 )− y(i)
1 )· x(i)
1
θ0 := θtemp_0
θ1 := θtemp_1
• if α is too small, gradient descent can be slow
• if α is too large, gradient descent can overshoot the minimum. It may be fail to converge, or even diverge
Gradient descent alorithm works sucsessfully if after n iteration steps the partial derivative of the cost function is zero.
α·
∂
∂θ0
J(θ0,θ1) = α·0; α·
∂
∂θ1
J(θ0,θ1) = α·0 (5.2)
9
10. Supervised Learning Problem
Linear Regression / One Feature
In MATLAB we initialize the descent algorithm with our given input matrix X, given output matrix Y and an arbitrary
parameter vector θ. At the beginning we need to initialize the iteration steps iter and the learning rate alpha. To
check if our learning algorithm is working well we will calculate stepwise the value of our cost function. For this
reason we initialize a cost-converge-test function J_test. After the gradient descent algorithm is done we plot the
cost-converge-test function dependending on the iteration steps iters.
iter = 1000;
alpha = 0.01;
J_test = zeros(iter,1);
iters =[1:1:iter]’;
for k=1:iter
J_test(k) = 1/(2*m) * sum( (X*theta - Y).ˆ2 );
theta_temp1 = theta(1) - alpha*1/m * sum( X*theta - Y );
theta_temp2 = theta(2) - alpha*1/m * sum( (X*theta - Y ).*X(:,2) );
theta(1) = theta_temp1;
theta(2) = theta_temp2;
end
plot(k,J_test)
disp([’theta_0: ’, num2str(theta(1)),’ ; theta_1: ’,num2str(theta(2))])
10
11. Supervised Learning Problem
Linear Regression / One Feature
6 Normal Equation Algorithm (Alternative)
Gradient descent gives one way of minimizing J(θ0,θ1). Let’s discuss a second way of doing so, this time performing
the minimization explicitly and without resorting to an iterative algorithm. In the "Normal Equation" method, we will
minimize J(θ0,θ1) by explicitly taking its derivatives with respect to the θj ’s, and setting them to zero. This allows us
to find the optimum theta without iteration. The normal equation formula is given below:
θ = (X ·X)−1
·X ·Y (6.1)
With the normal equation, computing the inversion has complexity O(n3
). So if we have a very large number of
features, the normal equation will be slow. In practice, when n exceeds 10,000 it might be a good time to go from a
normal solution to an iterative process.
In MATLAB we write
theta = inv(X’*X)*X’*Y;
11
12. Supervised Learning Problem
Linear Regression / One Feature
7 Important Conclusions
• Gradient Descent Algorithm
– Need to choose α
– Needs many iterations O(kn2
)
– Works well when n is large
– if α is too small, gradient descent can be slow
– if α is too large, gradient descent can overshoot the minimum. It may be fail to converge, or even diverge
• Normal Equation Algorithm
– No need to choose α
– No need to iterate
– O(n3
), need to calculate inverse of (X · X)
– Slow if n is very large
– When implementing the normal equation in MATLAB we want to use the pinv function rather than inv.
The pinv function will give you a value of θ even if (X · X) is not invertible.
– If (X · X) is noninvertible, the common causes might be having
* Redundant features, where two features are very closely related (i.e. they are linearly dependent)
* Too many features (e.g. m ≤ n). In this case, delete some features or use regularization (see other
journal).
12
13. Supervised Learning Problem
Linear Regression / One Feature
Glossary
Ω Input set
Υ Output set
x0 Bias element
x1 Feature element
y1 Output element
u Arbitrary element in the input set
v Arbitrary element in the output set
m Number of feature examples
Π Training set
t(x(i)
1 ) Training function (discrete)
T Training matrix
X Input matrix
Y Output matrix
h(x1) Hypothesis function (continuous)
θ0 1-st element of the parameter vector
θ1 2-nd element of the parameter vector
θj j-th element of the parameter vector
θ Parameter vector
x Hypothesis-input vector
13
14. Supervised Learning Problem
Linear Regression / One Feature
J(θ0,θ1) Cost function
x(i)
1 i-th feature value
y(i)
1 i-th output value
α Learning rate
∂
∂θj
J(θ0,θ1) Partial derivative of the cost function
X Transpose of the input matrix
n Number of feature elements
14