Pattern Classification by Iteratively Determined Linear and Piecewise Linear Discriminant Functions

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

Nonlinear effects of a power amplifier (PA) in communications systems include spectral spreading, which causes adjacent channel interference. For predistorter design, it is crucial to employ a model that can sufficiently model the intermodulation distortion (IMD) caused by the PA. The order of IMD generated by a polynomial is determined by its highest order basis. For a piecewise linear function, the order of nonlinearity cannot be directly deduced from its basis function. This paper investigates how IMD is introduced by the simplicial canonical piecewise linear (SCPWL) function. We examine the operation imposed by the SCPWL basis function on sinusoid and multi-tone signal in the time domain and frequency domain. Our analysis results in expressions that can be used to predict the output spectrum of the SCPWL basis. The SCPWL basis operation involves spreading of the input signal spectrum. The output spectrum of each SCPWL basis includes infinite IMD components. The IMD order of the SCPWL model is not determined by the highest order basis. Instead, the number of SCPWL parameters, i.e., the number of linear affine segments that defines the function, determines how accurately it can model the IMD components.

Topology design of remote patient monitoring system concerning qualitative and quantitative issues

Purpose To determine and validate the diagnostic ability of a linear discriminant function(LDF) based on the retinal nerve fiber layer thickness at each of the 12 clockhour positions obtained using optical coherence tomography for discriminating between healthy eyes and eyes with early glaucomatous visual field loss Methods We prospectively selected 62 consecutive healthy individuals and 73 patients with open‐angle glaucoma to calculate the LDF. Another independent prospective sample of 280 healthy eyes and 302 glaucomatous eyes was used to evaluate the diagnostic accuracy of the LDF Results The proposed function was LDF=15.584 – (12‐ o’clock segment thickness_0.032) – (7‐o’clock segment thickness_0.041) – (3‐o’clock segment thickness[nasal side]_0.121). The greatest area under the receiver operating characteristic curve was observed for our LDF in both populations: 0.962 and 0.922. Our LDF and the average thickness yielded sensitivities of 74.5% and 67.8%, respectively, at a fixed specificity of 95% Conclusion The LDF increased the diagnostic ability of the isolated retinal nerve fiber layer thickness at the 12 clock‐hour positions. Compared with optical coherence tomography–provided parameters, our LDF had the highest sensitivities at 85% and 95% fixed specificities to discriminate between healthy and early glaucomatous eyes

To determine and validate the diagnostic ability of a linear discriminant function (LDF) based on the retinal nerve fiber layer thickness at each of the 12 clock-hour positions obtained using optical coherence tomography for discriminating between healthy eyes and eyes with early glaucomatous visual field loss. We prospectively selected 62 consecutive healthy individuals and 73 patients with open-angle glaucoma to calculate the LDF. Another independent prospective sample of 280 healthy eyes and 302 glaucomatous eyes was used to evaluate the diagnostic accuracy of the LDF. The proposed function was LDF = 15.584 - (12-o'clock segment thickness x 0.032) - (7-o'clock segment thickness x 0.041) - (3-o'clock segment thickness [nasal side] x 0.121). The greatest area under the receiver operating characteristic curve was observed for our LDF in both populations: 0.962 and 0.922. Our LDF and the average thickness yielded sensitivities of 74.5% and 67.8%, respectively, at a fixed specificity of 95%. The LDF increased the diagnostic ability of the isolated retinal nerve fiber layer thickness at the 12 clock-hour positions. Compared with optical coherence tomography-provided parameters, our LDF had the highest sensitivities at 85% and 95% fixed specificities to discriminate between healthy and early glaucomatous eyes.

The stability of decision—Theoretic electrocardiographic classifiers based on the use of discretized features

In this study, seismic events in Northern and Central Egypt are inspected to discriminate quarry blasts from earthquakes. We examine a collection of 639 events in both time and frequency domains with local magnitudes of 1.5 ≤ ML ≤ 3.3 from the Egyptian Seismological Network’s seismic event catalogue between 2009 and 2015. The maximum S-wave to the maximum P-wave amplitude ratio, complexity (C), spectral ratio (Sr), and power of events (Pe) classifiers as well as two statistical approaches, linear discriminant function (LDF) and quadratic discriminant function (QDF), are used to distinguish between earthquakes and quarry blasts. The usage of the LDF and QDF forms did not result in any major differences in the discrimination. The results obtained by the LDF and QDF from (Pe-C) are the best of all approaches. The findings of all approaches were compared to get a final categorization for each event, and a decision was achieved when at least three of the four methods provided the same event category. In Northern Egypt, 243 earthquakes and 308 quarry blasts could be identified as final decisions, with two misclassified events, resulting in an overall success rate of 99.6%. In Central Egypt, 48 earthquakes and 36 quarry blasts were classified as a final decision, with two misclassified events, for an overall success percentage of 97.6%.

Fisher proposes Fisher’s linear discriminant function (LDF) based on maximization of variance ratio. If data satisfies Fisher’s assumption, the same LDF is easily derived by variance-covariance matrix. If variance-covariance matrices of two classes are not same, quadratic discriminant function (QDF) is proposed. These discriminant functions have three problems. First, all discriminant functions except for Revised IP-OLDF can’t discriminate the cases xi on the discriminant hyper plane correctly (the unresolved problem of discriminant analysis). Only Revised IP-OLDF can solve this problem in theoretically. Second, LDF and QDF can’t recognize linear separable data in many cases, and the numbers of misclassifications (NMs) of these functions are usually higher than logistic regression that is used in many users. Third, these are not obtained if the values of some independent variable are constant, because inverse matrix can’t be calculated. JMP prepare the regularized discriminant function if QDF can’t analyze the difficult data. These facts mean that LDF and QDF shouldn’t be used for an important discrimination. On the contrary, Revised IP-OLDF based on Minimum Number of Misclassifications (MNM) criterion resolves these problems. In addition to this, means of error rates of Revised IP-OLDF are less than those of LDF and logistic regression and S-SVM by 100-fold cross validation in many cases.

The purpose of the study was to create a linear discriminant function (LDF) formula by using the new Glaucoma Probability Score (GPS) global and sectorial optic nerve head parameters measured by Heidelberg Retinal Tomograph 3 to improve the GPS diagnostic capacity to discriminate between healthy and glaucomatous eyes. This is a multicenter cross-sectional study. To calculate the LDF formula, 137 normal individuals and 96 glaucomatous patients were selected. Another independent sample of 60 healthy and 69 glaucomatous eyes was used to evaluate the diagnostic accuracy of the LDF formulas. All patients underwent a full eye examination, standard achromatic perimetry by using Humphrey Field Analyzer, and imaging with Heidelberg Retinal Tomograph 3. Glaucoma was defined on the basis of SITA-24-2 visual field loss (pattern standard deviation P<5% and glaucoma Hemifield test outside normal limits). The area under the receiver operating characteristics curves and sensitivity and specificity for different LDFs were analyzed as measures of diagnostic accuracy. The analysis was repeated after stratification for optic disc size and glaucoma stage. Two LDF formulas improved GPS algorithm results. At fixed specificities of 85% and 95%, the sensitivity values were 79.7% and 71%, respectively, for the GPS-only LDF, which used only the so-called GPS parameters, whereas the values were 85.5% and 79.7%, respectively, for the GPS-RA LDF, which used sectorial rim area parameters as well. When the so-called GPS parameters were used in an LDF formula, its diagnostic capacity slightly improved; however, the diagnostic performances of the GPS LDF, which used sectorial rim area parameters as well, were better.

This paper proposes a new mathematical programming approach to represent the dimensions of the discriminant space for the multiple-group classification problem. Few papers have investigated generalizations of two-group mathematical programming approaches for the classification of multiple groups. While several papers have proposed mathematical programming models for separating groups of observations, the issue of considering the classification problem by finding discriminant linear functions to describe the groups in fewer dimensions has not been addressed. The new mathematical programming approach proposed in this paper first solves the multiple-group problem using a single discriminant function, which essentially represents the separation of the groups in one dimension. Then the multiple-group problem is successively solved using single discriminant functions with the requirement that successive linear discriminant functions have a sample covariance equal to zero. An algorithm is proposed to classify observations from multiple groups using the linear discriminant functions from the mathematical programming approach in a reduced number of dimensions.

This paper describes an accurate and efficient method for supervised classification of multispectral images. First, a simple derivation of a best linear discriminant function (BLD) is presented through geometrical consideration on ellipses with equi-Mahalanobis distance. It is shown that the function satisfies the Minimax criterion, which implies the robustness with regard to prior probabilities. Then, the binary decision tree (BDT) is introduced in order to make the method efficient, where the BLD is utilized as a decision rule. Care is taken to prevent a decline in the classification accuracy during the process of constructing the BDT. In addition to the theoretical evaluation of the processing speed, the actual performance of the proposed method is shown by applying it to multispectral data of Landsat-5. It is shown that the BDT/BLD becomes more efficient than Fisher's linear discriminant function when the number of classes increases.

Revised IP-OLDF (optimal linear discriminant function by integer programming) is a linear discriminant function to minimize the number of misclassifications (NM) of training samples by integer programming (IP). However, IP requires large computation (CPU) time. In this paper, it is proposed how to reduce CPU time by using linear programming (LP). In the first phase, Revised LP-OLDF is applied to all cases, and all cases are categorized into two groups: those that are classified correctly or those that are not classified by support vectors (SVs). In the second phase, Revised IP-OLDF is applied to the misclassified cases by SVs. This method is called Revised IPLP-OLDF.In this research, it is evaluated whether NM of Revised IPLP-OLDF is good estimate of the minimum number of misclassifications (MNM) by Revised IP-OLDF. Four kinds of the real data—Iris data, Swiss bank note data, student data, and CPD data—are used as training samples. Four kinds of 20,000 re-sampling cases generated from these data are used as the evaluation samples. There are a total of 149 models of all combinations of independent variables by these data. NMs and CPU times of the 149 models are compared with Revised IPLP-OLDF and Revised IP-OLDF. The following results are obtained: 1) Revised IPLP-OLDF significantly improves CPU time. 2) In the case of training samples, all 149 NMs of Revised IPLP-OLDF are equal to the MNM of Revised IP-OLDF. 3) In the case of evaluation samples, most NMs of Revised IPLP-OLDF are equal to NM of Revised IP-OLDF. 4) Generalization abilities of both discriminant functions are concluded to be high, because the difference between the error rates of training and evaluation samples are almost within 2%. Therefore, Revised IPLP-OLDF is recommended for the analysis of big data instead of Revised IP-OLDF. Next, Revised IPLP-OLDF is compared with LDF and logistic regression by 100-fold cross validation using 100 re-sampling samples. Means of error rates of Revised IPLP-OLDF are remarkable fewer than those of LDF and logistic regression.

Species identification is a fundamental process for ecological studies and conservation practices, and simple nonlethal identification criteria are important for endangered species. This study developed species identification criteria for two endangered freshwater pearl mussels ( Margaritifera laevis and Margaritifera togakushiensis ) based on linear discriminant functions (LDFs) that were established considering intraspecific regional morphological differences from sites at Hokkaido, northern Japan. We collected a total of 1,110 mussels from 52 rivers across 32 watersheds in two geographical regions (east and west). Shell morphologies (length, height, and width) of the collected mussels were measured, followed by species identification with gel electrophoresis banding patterns of 16S rRNA polymerase chain reaction products. LDFs were constructed for two size classes (≥40 mm and &lt; 40 mm) using forward stepwise procedures to determine key morphological differences between the two species and consideration of whether regional morphological differences improved identification accuracies. The LDFs revealed that the morphologies of the two species were clearly distinguished by the relationship between height and length for both size classes. Region‐specific LDFs produced better identification accuracies. Regardless of region, the maximum length of M. laevis exceeded 100 mm, whereas all M. togakushiensis were &lt; 100 mm in length. Identification accuracies of the established LDFs were high for each of the five length classes (0–20 mm, 20–40 mm, 40–60 mm, 60–80 mm, and 80–100 mm) with 85–96% (mean: 92%) accuracy in the east and 67–96% (mean: 80%) in the west. These criteria for species identification will progress future ecological studies and conservation practices for freshwater mussels.

This study describes the derivation and validation of the early detection of alcohol consumption (EDAC) test, which uses linear discriminant function (LDF) analysis for the identification of alcohol misuse. This form of LDF aims to predict a categorical dependent variable (alcohol misuse) on the basis of several independent, predictor variables (routine laboratory tests). EDAC was developed to classify individuals as heavy or light drinkers using a database of 1599 subjects recruited from 25 sites in the USA. The predictor variables for the LDF were 36 routine chemistry and haematology analytes. The EDAC model produced 80.7% sensitivity and 84.4% specificity, with an overall correct classification rate of 82.5%. Using a stepwise method, 20 of the 36 routine tests used in the LDF were selected as the optimal predictor variables. The top three variables with the highest contribution in the stepwise EDAC model were: bilirubin ratio (direct to total), aspartate aminotransferase and albumin. This study shows that LDF analysis in conjunction with new, user-friendly computer packages is a practical and cost-effective laboratory tool for detecting excessive drinking using blood constituents ordered routinely in a variety of clinical settings. Diagnostic performance can be adjusted to achieve higher specificity rates.