DOCSLIB.ORG

Application of a Logistic Function to Describe the Growth of Fodder Galega

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Application of a Logistic Function to Describe the Growth of Fodder Galega Journal of Ecological Engineering Volume 18, Issue 1, Jan. 2017, pages 125–131 DOI: 10.12911/22998993/66245 Research Article APPLICATION OF A LOGISTIC FUNCTION TO DESCRIBE THE GROWTH OF FODDER GALEGA Katarzyna Rymuza1, Antoni Bombik1 1 University of Natural Sciences and Humanities in Siedlce, Prusa 14, 08-110 Siedlce, Poland, e-mail: katarzyna. [email protected] Received: 2016.07.20 ABSTRACT Accepted: 2016.10.23 It was attempted in the paper to describe the growth of oriental goat’s rue (Galega Published: 2017.01.01 orientalis Lam.) growth by means of a logistic function. The studies demonstrated that the function describes empirical data well as the coefficients of determination obtained ranged between 97 to 98%, depending on the growing season. The estimated function parameters indicate that both the plant height and growth rate were condi- tioned by thermal conditions and moisture during the growing season. Plants had the longest stems in the year 2010 (127.9 cm), and the shortest in 2008 (105.9 cm). The maximum growth rate ranged from 2.12 to 3.97 cm per day in, respectively, 2009 and 2008. Also the inflection point, the point at which plants grew the fastest, depended on the conditions of the growing season. Keywords: oriental goat’s rue, logistic function, growth equation, growth rate equa- tion, growth rate. INTRODUCTION goat’s rue, conducted by Raig [2001], Ignac- zak [2010] have demonstrated that the develop- Oriental goat’s rue (Galega orientalis Lam.) ment of the plant is a complex process. During is a plant of the family Fabaceae Lindl., subfam- the growing season, leaf rosettes are formed on ily Fabaoideae, tribe Galegae and subtribe Gale- root necks in early spring. When the minimum ginae. It is only recently that scientists world- temperature exceeds 5oC, plants start to produce wide [Šlepetys, 2003; Baležentienė, 2009, 2011; aerial stems from root necks that have overwin- Egamberdieva et al., 2010; Mikkonen et al., 2011] tered, and one-year-old underground roots. Dur- and in Poland [Deska, 2009; Deska et al., 2012, ing blossoming and pod maturation, cluster tips Kozłowski and Zielewicz, 2013] have started to wither and shrink if rainfall is insufficient. Under be interested in the plant. Indigenous to the north- the climatic conditions of Poland, pods in main ern Caucasus, the plant can be found in the wild clusters mature in late July. However, if late July in forested regions of Georgia, southern Arme- is rainy, oriental goat’s rue’s plants develop sec- nia, south-western Azerbaijan and sometimes in ondary and tertiary stems which produce flowers the Crimea. Due to the fact that the plant’s green [Deska and Wyrzykowska, 2005; Deska, 2009; parts are rich in protein (the content ranges be- Deska et al., 2012]. tween 16–27%), oriental goat’s rue is commonly Similarly to any other plant, the development grown in some EU Baltic countries (Estonia, Fin- of oriental goat’s rue can be described mathe- land, Latvia) and in Russia. In Poland the species matically by means of indices and functions. The is relatively unpopular, just like in France, Can- functions that describe the relationship: age-time, ada, Croatia, Belarus and Serbia where the first are called growth curves and make it possible to attempts are being made to introduce the plant on include many measurements into one equation a wider scale. Studies on the biology of growth based on several parameters which can be in- and morphological development of the oriental terpreted from a biological standpoint. They are 125 Journal of Ecological Engineering Vol. 18(1), 2017 sigmoid curves consisting of three stages: “expo- MATERIALS AND METHODS nental” characterized by slow but constantly in- creasing growth, “linear” when the growth is the Analysis of data was based on measurements fastest till maximum is reached, and the stage of of stem lengths of oriental goat’s rue (Galega ori- “growing old” when the growth slows [Gregorc- entalis Lam.) cv. Gale which were taken during zyk, 1991; Wesołowska-Janczarek, 1993; Kara- a four-year period as part of a long-term experi- davut et al., 2008]. Analysis of plant growth fre- ment set up in 2007 at the University of Natu- quently employs the logistic function – a function ral Sciences and Humanities experimental site which is continuous and differentiable at each (52º10’N, 22º17’E Poland). point in its domain, with an inflection point with From 2007 to 2010, 70 plants were randomly respect to which the graph is symmetrical [Tsou- selected every seven days from the start of growth laris, 2001; Tsoularis and Wallace, 2002]. to the stage of full maturity of pods. The begin- The mathematical description of a completed ning of growth was assumed to be the moment process may contribute to enhanced knowledge of forming leaf buds on root necks of the plants of the biology of plant development, indicate which overwintered. Thermal conditions and moments of optimum agrotechnological actions, rainfall during the growing seasons are presented help to forecast yields as well as indicate mor- in Figures 1 and 2. phological and developmental differences of in- The temperature varied in all the study years. dividuals and the whole population. Relatively cool spring of 2007 saw a sudden in- The objective of the work was to examine crease in temperature in late April, and high tem- the growth of oriental goat’s rue Galega orien- peratures at the end of the growing season. In the talis Lam.) during a 4-year period by means years 2008 and 2010 the beginning of the grow- of a logistic function. ing season was characterized by quite low tem- 2007 2008 2009 2010 1987 - 2010 30,0 25,0 20,0 C) o 15,0 10,0 5,0 temperature ( temperature 0,0 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 -5,0 III IV V VI VII months and ten-day temperature Figure 1. Ten-day and monthly temperatures (oC) in the study years 80,0 2007 2008 2009 2010 1987 - 2010 70,0 60,0 50,0 40,0 30,0 20,0 10,0 precipitation (mm) 0,0 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 III IV V VI VII months and ten-day periods Figure 2. Ten day and monthly precipitation (mm) in the study years 126 Journal of Ecological Engineering Vol. 18(1), 2017 peratures, around 5oC, but later they increased Moreover, the equation of growth rate was steadily. The temperature in the year 2009 was calculated in addition to the theoretical initial the lowest in the spring and was still around 1oC value of growth and inflection points, according as late as in early April. Then a sudden increase in to the formulae [Gregorczyk, 1991]: temperature was observed. • equation of growth rate: The worst hydrological conditions were re- dy k corded in 2007 and 2009 when the precipitation y(t)(a y(t)) dt a did not exceed 200 cm during the growing sea- son. Moreover, it was unevenly distributed over • theoretical initial value of growth: time. With this respect, the best year was 2010 a y0 when the high rainfall was also evenly distributed 1 b and, as a result, met the needs of plants. • coordinates of the inflection point of the The following function was used to describe function: the growth of oriental goat’s rue [Richards, 1969]: ln b a y(t) = t i , y(ti ) 0,5 a 1+ b ⋅ e −k⋅t k where: y(t) – denotes plant’s height on day t, In order to compare the growth patterns in a – the value of asymptote, individual years, the function was linearized fol- b – an integration constant and lowing the formula: lnb – k×t. Next, the hypothe- k – the index of growth rate. sis was checked whether or not the slopes of strait The terms a, k and b are somehow interdepend- lines are equal so as to find out if the course of the straight lines examined is the same. The hypoth- able. If k values are high, a values are low and de- = = = = scribe small plants. By contrast, low k values gener- esis H0 k1 k2 k3 k4 was checked using ate high values, which is typical of high plants. Val- Tukey’s test following Elandt [1964]. ues of the parameter b depend on initial values and, All the calculations were performed using in general, higher plants have got lower b values. Statistica 9.0 and the Excel spreadsheet. Equations of functions of average growth were estimated for each year separately. The cri- RESULTS terion of goodness of fit was the determination 2 coefficient R , which is frequently used for this Figure 1 shows the growth of goat’s rue in the purpose, calculated following the formula [Seber study years. It demonstrates that the growth was and Wild, 1989; Ryan, 1997]: the best in 2010 when precipitation distribution SSE R 2 = 1− coincided with the water needs of plants. Also SST function parameters, describing growth equations in which: and growth rate equations, confirm this finding n n 2 (Table 1, 2 and Figure 3). In 2010 the plants had SSE = (y − f ) 2 , SST = (y − y) , ∑ i i ∑ i the highest theoretical asymptotic value (param- i=1 i=1 eter a) and low values of coefficients k and b. where: y – experimental data; i In 2010 oriental goat’s rue plants as early f – values of the function fitted corre- i as after 42 days of growth reached the inflec- sponding to the values y , i tion points at the stem length of 64 cm and the y – arithmetical mean of the experimen- maximum growth rate of 3.45 cm ·day-1.
Load more

Recommended publications
  • 6.5 Applications of Exponential and Logarithmic Functions 469
    6.5 Applications of Exponential and Logarithmic Functions 469 6.5 Applications of Exponential and Logarithmic Functions As we mentioned in Section 6.1, exponential and logarithmic functions are used to model a wide variety of behaviors in the real world. In the examples that follow, note that while the applications are drawn from many different disciplines, the mathematics remains essentially the same. Due to the applied nature of the problems we will examine in this section, the calculator is often used to express our answers as decimal approximations. 6.5.1 Applications of Exponential Functions Perhaps the most well-known application of exponential functions comes from the financial world. Suppose you have $100 to invest at your local bank and they are offering a whopping 5 % annual percentage interest rate. This means that after one year, the bank will pay you 5% of that $100, or $100(0:05) = $5 in interest, so you now have $105.1 This is in accordance with the formula for simple interest which you have undoubtedly run across at some point before. Equation 6.1. Simple Interest The amount of interest I accrued at an annual rate r on an investmenta P after t years is I = P rt The amount A in the account after t years is given by A = P + I = P + P rt = P (1 + rt) aCalled the principal Suppose, however, that six months into the year, you hear of a better deal at a rival bank.2 Naturally, you withdraw your money and try to invest it at the higher rate there.
    [Show full text]
  • Chapter 10. Logistic Models
    www.ck12.org Chapter 10. Logistic Models CHAPTER 10 Logistic Models Chapter Outline 10.1 LOGISTIC GROWTH MODEL 10.2 CONDITIONS FAVORING LOGISTIC GROWTH 10.3 REFERENCES 207 10.1. Logistic Growth Model www.ck12.org 10.1 Logistic Growth Model Here you will explore the graph and equation of the logistic function. Learning Objectives • Recognize logistic functions. • Identify the carrying capacity and inflection point of a logistic function. Logistic Models Exponential growth increases without bound. This is reasonable for some situations; however, for populations there is usually some type of upper bound. This can be caused by limitations on food, space or other scarce resources. The effect of this limiting upper bound is a curve that grows exponentially at first and then slows down and hardly grows at all. This is characteristic of a logistic growth model. The logistic equation is of the form: C C f (t) = 1+ab−t = 1+ae−kt The above equations represent the logistic function, and it contains three important pieces: C, a, and b. C determines that maximum value of the function, also known as the carrying capacity. C is represented by the dashed line in the graph below. 208 www.ck12.org Chapter 10. Logistic Models The constant of a in the logistic function is used much like a in the exponential function: it helps determine the value of the function at t=0. Specifically: C f (0) = 1+a The constant of b also follows a similar concept to the exponential function: it helps dictate the rate of change at the beginning and the end of the function.
    [Show full text]
  • Neural Network Architectures and Activation Functions: a Gaussian Process Approach
    Fakultät für Informatik Neural Network Architectures and Activation Functions: A Gaussian Process Approach Sebastian Urban Vollständiger Abdruck der von der Fakultät für Informatik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. rer. nat. Stephan Günnemann Prüfende der Dissertation: 1. Prof. Dr. Patrick van der Smagt 2. Prof. Dr. rer. nat. Daniel Cremers 3. Prof. Dr. Bernd Bischl, Ludwig-Maximilians-Universität München Die Dissertation wurde am 22.11.2017 bei der Technischen Universtät München eingereicht und durch die Fakultät für Informatik am 14.05.2018 angenommen. Neural Network Architectures and Activation Functions: A Gaussian Process Approach Sebastian Urban Technical University Munich 2017 ii Abstract The success of applying neural networks crucially depends on the network architecture being appropriate for the task. Determining the right architecture is a computationally intensive process, requiring many trials with different candidate architectures. We show that the neural activation function, if allowed to individually change for each neuron, can implicitly control many aspects of the network architecture, such as effective number of layers, effective number of neurons in a layer, skip connections and whether a neuron is additive or multiplicative. Motivated by this observation we propose stochastic, non-parametric activation functions that are fully learnable and individual to each neuron. Complexity and the risk of overfitting are controlled by placing a Gaussian process prior over these functions. The result is the Gaussian process neuron, a probabilistic unit that can be used as the basic building block for probabilistic graphical models that resemble the structure of neural networks.
    [Show full text]
  • Neural Networks
    School of Computer Science 10-601B Introduction to Machine Learning Neural Networks Readings: Matt Gormley Bishop Ch. 5 Lecture 15 Murphy Ch. 16.5, Ch. 28 October 19, 2016 Mitchell Ch. 4 1 Reminders 2 Outline • Logistic Regression (Recap) • Neural Networks • Backpropagation 3 RECALL: LOGISTIC REGRESSION 4 Using gradient ascent for linear Recall… classifiers Key idea behind today’s lecture: 1. Define a linear classifier (logistic regression) 2. Define an objective function (likelihood) 3. Optimize it with gradient descent to learn parameters 4. Predict the class with highest probability under the model 5 Using gradient ascent for linear Recall… classifiers This decision function isn’t Use a differentiable function differentiable: instead: 1 T pθ(y =1t)= h(t)=sign(θ t) | 1+2tT( θT t) − sign(x) 1 logistic(u) ≡ −u 1+ e 6 Using gradient ascent for linear Recall… classifiers This decision function isn’t Use a differentiable function differentiable: instead: 1 T pθ(y =1t)= h(t)=sign(θ t) | 1+2tT( θT t) − sign(x) 1 logistic(u) ≡ −u 1+ e 7 Recall… Logistic Regression Data: Inputs are continuous vectors of length K. Outputs are discrete. (i) (i) N K = t ,y where t R and y 0, 1 D { }i=1 ∈ ∈ { } Model: Logistic function applied to dot product of parameters with input vector. 1 pθ(y =1t)= | 1+2tT( θT t) − Learning: finds the parameters that minimize some objective function. θ∗ = argmin J(θ) θ Prediction: Output is the most probable class. yˆ = `;Kt pθ(y t) y 0,1 | ∈{ } 8 NEURAL NETWORKS 9 Learning highly non-linear functions f: X à Y l f might be non-linear
    [Show full text]
  • Lecture 5: Logistic Regression 1 MLE Derivation
    CSCI 5525 Machine Learning Fall 2019 Lecture 5: Logistic Regression Feb 10 2020 Lecturer: Steven Wu Scribe: Steven Wu Last lecture, we give several convex surrogate loss functions to replace the zero-one loss func- tion, which is NP-hard to optimize. Now let us look into one of the examples, logistic loss: given d parameter w and example (xi; yi) 2 R × {±1g, the logistic loss of w on example (xi; yi) is defined as | ln (1 + exp(−yiw xi)) This loss function is used in logistic regression. We will introduce the statistical model behind logistic regression, and show that the ERM problem for logistic regression is the same as the relevant maximum likelihood estimation (MLE) problem. 1 MLE Derivation For this derivation it is more convenient to have Y = f0; 1g. Note that for any label yi 2 f0; 1g, we also have the “signed” version of the label 2yi − 1 2 {−1; 1g. Recall that in general su- pervised learning setting, the learner receive examples (x1; y1);:::; (xn; yn) drawn iid from some distribution P over labeled examples. We will make the following parametric assumption on P : | yi j xi ∼ Bern(σ(w xi)) where Bern denotes the Bernoulli distribution, and σ is the logistic function defined as follows 1 exp(z) σ(z) = = 1 + exp(−z) 1 + exp(z) See Figure 1 for a visualization of the logistic function. In general, the logistic function is a useful function to convert real values into probabilities (in the range of (0; 1)). If w|x increases, then σ(w|x) also increases, and so does the probability of Y = 1.
    [Show full text]
  • An Introduction to Logistic Regression: from Basic Concepts to Interpretation with Particular Attention to Nursing Domain
    J Korean Acad Nurs Vol.43 No.2, 154 -164 J Korean Acad Nurs Vol.43 No.2 April 2013 http://dx.doi.org/10.4040/jkan.2013.43.2.154 An Introduction to Logistic Regression: From Basic Concepts to Interpretation with Particular Attention to Nursing Domain Park, Hyeoun-Ae College of Nursing and System Biomedical Informatics National Core Research Center, Seoul National University, Seoul, Korea Purpose: The purpose of this article is twofold: 1) introducing logistic regression (LR), a multivariable method for modeling the relationship between multiple independent variables and a categorical dependent variable, and 2) examining use and reporting of LR in the nursing literature. Methods: Text books on LR and research articles employing LR as main statistical analysis were reviewed. Twenty-three articles published between 2010 and 2011 in the Journal of Korean Academy of Nursing were analyzed for proper use and reporting of LR models. Results: Logistic regression from basic concepts such as odds, odds ratio, logit transformation and logistic curve, assumption, fitting, reporting and interpreting to cautions were presented. Substantial short- comings were found in both use of LR and reporting of results. For many studies, sample size was not sufficiently large to call into question the accuracy of the regression model. Additionally, only one study reported validation analysis. Conclusion: Nurs- ing researchers need to pay greater attention to guidelines concerning the use and reporting of LR models. Key words: Logit function, Maximum likelihood estimation, Odds, Odds ratio, Wald test INTRODUCTION The model serves two purposes: (1) it can predict the value of the depen- dent variable for new values of the independent variables, and (2) it can Multivariable methods of statistical analysis commonly appear in help describe the relative contribution of each independent variable to general health science literature (Bagley, White, & Golomb, 2001).
    [Show full text]
  • Fm 3-34 Engineer Operations Headquarters, Department
    FM 3-34 April 2009 ENGINEER OPERATIONS DISTRIBUTION RESTRICTION: Approved for public release; distribution is unlimited. HEADQUARTERS, DEPARTMENT OF THE ARMY This publication is available at Army Knowledge Online (AKO) <www.us.army.mil> and the General Dennis J. Reimer Training and Doctrine Digital Library at <www.train.army.mil>. *FM 3-34 Field Manual No. Headquarters 3-34 Department of the Army Washington, DC, 2 April 2009 Engineer Operations Contents Page PREFACE .............................................................................................................. v INTRODUCTION .................................................................................................. vii Chapter 1 THE OPERATIONAL ENVIRONMENT ............................................................. 1-1 Understanding the Operational Environment ..................................................... 1-1 The Military Variable ........................................................................................... 1-4 Spectrum of Requirements ................................................................................. 1-5 Support Spanning the Levels of War.................................................................. 1-7 Engineer Soldiers ............................................................................................... 1-9 Chapter 2 ENGINEERING IN UNIFIED ACTION ............................................................... 2-1 SECTION I–THE ENGINEER REGIMENT ........................................................ 2-1 The
    [Show full text]
  • Notes: Logistic Functions I. Use, Function, and General Graph If
    Notes: Logistic Functions I. Use, Function, and General Graph If growth begins slowly, then increases rapidly and eventually levels off, the data often can be model by an “S-curve”, or a logistic function. The logistic function describes certain kinds of growth. These functions, like exponential functions, grow quickly at first, but because of restrictions that place limits on the size of the underlying population, eventually grow more slowly and then level off. For real numbers a, b, and c, the function is a logistic function. If a >0, a logistic function increases when b >0 and decreases when b < 0. Logistic Growth Function Equation: x-Intercept: none y-intercept: Horizontal Asymptote: The number, , is called the limiting value or the upper limit of the function because the graph of a logistic growth function will have a horizontal asymptote at y = c. As is clear from the graph above, the characteristic S-shape in the graph of a logistic function shows that initial exponential growth is followed by a period in which growth slows and then levels off, approaching (but never attaining) a maximum upper limit. Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time; in a different context, they can also describe demand curves: the decline of demand for a product as a function of increasing price can be modeled by a logistic function, as in the figure below.
    [Show full text]
  • Sigmoid Functions in Reliability Based Management 2007 15 2 67 2.1 the Nature of Performance Growth
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Periodica Polytechnica (Budapest University of Technology and Economics) Ŕ periodica polytechnica Sigmoid functions in reliability based Social and Management Sciences management 15/2 (2007) 67–72 doi: 10.3311/pp.so.2007-2.04 Tamás Jónás web: http://www.pp.bme.hu/so c Periodica Polytechnica 2007 RESEARCH ARTICLE Received 2008-09-14 Abstract 1 Introduction This paper introduces the theoretical background of a few A sigmoid function is a mathematical function that produces possible applications of the so-called sigmoid-type functions in a curve having an "S" shape, and is defined by the manufacturing and service management. 1 σλ,x (x) (1) An extended concept of reliability, derived from fuzzy theory, 0 λ(x x0) = 1 e− − is discussed here to illustrate how reliability based management + formula. The sigmoid function is also called sigmoidal curve decisions can be made consistent, when handling of weakly de- [1] or logistic function. The interpretation of sigmoid-type func- fined concepts is needed. tions – I use here – is that any function that can be transformed I demonstrate how performance growth can be modelled us- into σλ,x (x) through substitutions and linear transformations ing an aggregate approach with support of sigmoid-type func- 0 is considered as a sigmoid-type one. There are several well- tions. known applications of sigmoid-type functions. A few examples Suitably chosen parameters of sigmoid-type functions allow are: threshold function in neural networks [2], approximation these to be used as failure probability distribution and survival of Gaussian probability distribution, logistic regression [3], or functions.
    [Show full text]
  • Learning Activation Functions in Deep Neural Networks
    UNIVERSITÉ DE MONTRÉAL LEARNING ACTIVATION FUNCTIONS IN DEEP NEURAL NETWORKS FARNOUSH FARHADI DÉPARTEMENT DE MATHÉMATIQUES ET DE GÉNIE INDUSTRIEL ÉCOLE POLYTECHNIQUE DE MONTRÉAL MÉMOIRE PRÉSENTÉ EN VUE DE L’OBTENTION DU DIPLÔME DE MAÎTRISE ÈS SCIENCES APPLIQUÉES (GÉNIE INDUSTRIEL) DÉCEMBRE 2017 c Farnoush Farhadi, 2017. UNIVERSITÉ DE MONTRÉAL ÉCOLE POLYTECHNIQUE DE MONTRÉAL Ce mémoire intitulé : LEARNING ACTIVATION FUNCTIONS IN DEEP NEURAL NETWORKS présenté par : FARHADI Farnoush en vue de l’obtention du diplôme de : Maîtrise ès sciences appliquées a été dûment accepté par le jury d’examen constitué de : M. ADJENGUE Luc-Désiré, Ph. D., président M. LODI Andrea, Ph. D., membre et directeur de recherche M. PARTOVI NIA Vahid, Doctorat, membre et codirecteur de recherche M. CHARLIN Laurent, Ph. D., membre iii DEDICATION This thesis is dedicated to my beloved parents, Ahmadreza and Sholeh, who are my first teachers and always love me unconditionally. This work is also dedicated to my love, Arash, who has been a great source of motivation and encouragement during the challenges of graduate studies and life. iv ACKNOWLEDGEMENTS I would like to express my gratitude to Prof. Andrea Lodi for his permission to be my supervisor at Ecole Polytechnique de Montreal and more importantly for his enthusiastic encouragements and invaluable continuous support during my research and education. I am very appreciated to him for introducing me to a MITACS internship in which I have developed myself both academically and professionally. I would express my deepest thanks to Dr. Vahid Partovi Nia, my co-supervisor at Ecole Poly- technique de Montreal, for his supportive and careful guidance and taking part in important decisions which were extremely precious for my research both theoretically and practically.
    [Show full text]
  • Applied Machine Learning
    Applied Machine Learning Annalisa Marsico OWL RNA Bionformatics group Max Planck Institute for Molecular Genetics Free University of Berlin 13 Mai, SoSe 2015 Neural Networks Motivation: the non-linear hypothesis x x x Goal: learn the non-linear problem x x x x x One solution: add a lot of non-linear features x as input to the logistic function x x x x x x x x x = ( + + + + x + + + ⋯ . ) x x g = sigmoid function If more than two dimensions, including polynomial feature not a good idea It might lead to over fitting. Neural Networks • Solution : SVM with projection in a high-dimensional space. Kernel ‘trick’. • New solution: Neural Networks to learn non-linear complex hypothesis Neural Networks (NNs) – Biological motivation • Originally motivated by the desire of having machines which can mimic the brain • Widely used in the 80s and early 90s, popularity diminished in the late 90s • Recent resurgence: more computational power to run large-scale NNs The ‘brain’ algorithm • Fascinating hypothesis (Roe et al. 1992): the brain functions through a single learning algorithm Auditory cortex X Neurons Schematic representation of a single neuron in the brain - Neuron is a computational unit ‘Inputs’ (dendrites) Synapse Node of Cell body Ranvier Schwann cell Myelin sheath Cell nucleus Communication between neurons Modeling neurons Logistic unit Activation function is often a sigmoid 1 ≡ ℎ = ≡ () 1 + There exist other activation function: step function, linear function Logistic unit representation Bias unit, =1 () Sigmoid
    [Show full text]
  • Logistic Regression
    CHAPTER 5 Logistic regression Logistic regression is the standard way to model binary outcomes (that is, data yi that take on the values 0 or 1). Section 5.1 introduces logistic regression in a simple example with one predictor, then for most of the rest of the chapter we work through an extended example with multiple predictors and interactions. 5.1 Logistic regression with a single predictor Example: modeling political preference given income Conservative parties generally receive more support among voters with higher in- comes. We illustrate classical logistic regression with a simple analysis of this pat- tern from the National Election Study in 1992. For each respondent i in this poll, we label yi =1ifheorshepreferredGeorgeBush(theRepublicancandidatefor president) or 0 if he or she preferred Bill Clinton (the Democratic candidate), for now excluding respondents who preferred Ross Perot or other candidates, or had no opinion. We predict preferences given the respondent’s income level, which is characterized on a five-point scale.1 The data are shown as (jittered) dots in Figure 5.1, along with the fitted logistic regression line, a curve that is constrained to lie between 0 and 1. We interpret the line as the probability that y =1givenx—in mathematical notation, Pr(y =1x). We fit and display the logistic regression using the following R function calls:| fit.1 <- glm (vote ~ income, family=binomial(link="logit")) Rcode display (fit.1) to yield coef.est coef.se Routput (Intercept) -1.40 0.19 income 0.33 0.06 n=1179,k=2 residual deviance = 1556.9, null deviance = 1591.2 (difference = 34.3) 1 The fitted model is Pr(yi =1)=logit− ( 1.40 + 0.33 income).
    [Show full text]