This document discusses string primitive instructions in x86 assembly language. It provides an overview of instructions like MOVS, CMPS, SCAS, STOS, and LODS that are used to move, compare, scan, store, and load strings. It also describes how the direction flag affects the increment/decrement of the SI and DI registers. Procedures for comparing, copying, finding the length of, reading, writing, and converting case of strings are also summarized.
1) The document discusses parallel adders and subtractors for n-bit binary numbers. It specifically examines a 4-bit parallel adder that uses full adders connected in cascade, with the carry output of one full adder connected to the next's carry input.
2) A 4-bit parallel subtractor is also examined, which takes the 2's complement of the number to be subtracted and adds it to the other number using a 4-bit parallel adder.
3) Carry propagation time is discussed, which is the time it takes the carry to ripple through all the full adders in the parallel adder from the least to most significant bit.
Associative memory, also known as content-addressable memory (CAM), allows data to be searched based on its content rather than its location. It consists of a memory array, argument register (containing the search word), key register (specifying which bits to compare), and match register (indicating matching locations). All comparisons are done in parallel. Associative memory provides faster searching than conventional memory but is more expensive due to the additional comparison circuitry in each cell. It is well-suited for applications requiring very fast searching such as databases and virtual memory address translation.
The document discusses the minimum and maximum mode systems of the 8086 microprocessor. In minimum mode, the 8086 generates all control signals and a single processor is used. In maximum mode, an external bus controller chip generates control signals and multiple processors can be used. It describes the components, address latching, read and write cycles, and I/O interfacing for both minimum and maximum mode 8086 systems.
This document provides the lab manual for the IC Applications lab course for students in the III BTech ECE program. It includes an introduction, a list of 15 experiments to be performed in the lab divided into two parts, general do's and don'ts for the lab, and details on the first experiment - Adder, Subtractor, and Comparator using the IC 741 op-amp. The document provides theory, circuit diagrams, procedures, observation tables and model calculations for the first experiment.
Computer Organization And Architecture lab manualNitesh Dubey
The document discusses the implementation of various logic gates and flip-flops. It describes half adders and full adders can be implemented using XOR and AND gates. Binary to gray code and gray to binary code conversions are also explained. Circuit diagrams for 3-8 line decoder, 4x1 and 8x1 multiplexer are provided along with their truth tables. Finally, the working of common flip-flops like SR, JK, D and T are explained through their excitation tables.
The document discusses the concept of virtual memory. Virtual memory allows a program to access more memory than what is physically available in RAM by storing unused portions of the program on disk. When a program requests data that is not currently in RAM, it triggers a page fault that causes the needed page to be swapped from disk into RAM. This allows the illusion of more memory than physically available through swapping pages between RAM and disk as needed by the program during execution.
This document discusses string primitive instructions in x86 assembly language. It provides an overview of instructions like MOVS, CMPS, SCAS, STOS, and LODS that are used to move, compare, scan, store, and load strings. It also describes how the direction flag affects the increment/decrement of the SI and DI registers. Procedures for comparing, copying, finding the length of, reading, writing, and converting case of strings are also summarized.
1) The document discusses parallel adders and subtractors for n-bit binary numbers. It specifically examines a 4-bit parallel adder that uses full adders connected in cascade, with the carry output of one full adder connected to the next's carry input.
2) A 4-bit parallel subtractor is also examined, which takes the 2's complement of the number to be subtracted and adds it to the other number using a 4-bit parallel adder.
3) Carry propagation time is discussed, which is the time it takes the carry to ripple through all the full adders in the parallel adder from the least to most significant bit.
Associative memory, also known as content-addressable memory (CAM), allows data to be searched based on its content rather than its location. It consists of a memory array, argument register (containing the search word), key register (specifying which bits to compare), and match register (indicating matching locations). All comparisons are done in parallel. Associative memory provides faster searching than conventional memory but is more expensive due to the additional comparison circuitry in each cell. It is well-suited for applications requiring very fast searching such as databases and virtual memory address translation.
The document discusses the minimum and maximum mode systems of the 8086 microprocessor. In minimum mode, the 8086 generates all control signals and a single processor is used. In maximum mode, an external bus controller chip generates control signals and multiple processors can be used. It describes the components, address latching, read and write cycles, and I/O interfacing for both minimum and maximum mode 8086 systems.
This document provides the lab manual for the IC Applications lab course for students in the III BTech ECE program. It includes an introduction, a list of 15 experiments to be performed in the lab divided into two parts, general do's and don'ts for the lab, and details on the first experiment - Adder, Subtractor, and Comparator using the IC 741 op-amp. The document provides theory, circuit diagrams, procedures, observation tables and model calculations for the first experiment.
Computer Organization And Architecture lab manualNitesh Dubey
The document discusses the implementation of various logic gates and flip-flops. It describes half adders and full adders can be implemented using XOR and AND gates. Binary to gray code and gray to binary code conversions are also explained. Circuit diagrams for 3-8 line decoder, 4x1 and 8x1 multiplexer are provided along with their truth tables. Finally, the working of common flip-flops like SR, JK, D and T are explained through their excitation tables.
The document discusses the concept of virtual memory. Virtual memory allows a program to access more memory than what is physically available in RAM by storing unused portions of the program on disk. When a program requests data that is not currently in RAM, it triggers a page fault that causes the needed page to be swapped from disk into RAM. This allows the illusion of more memory than physically available through swapping pages between RAM and disk as needed by the program during execution.
The document contains a list of 23 microprocessor lab programs and 6 interfacing programs for an electronics and communication course. The programs cover topics like data transfer, arithmetic operations, sorting, prime number generation, string operations, matrix multiplication and more. The document provides contents, program descriptions and assembly language code for some of the programs.
A hardwired control unit generates control signals using finite state machines and a physically connected logic circuit. It consists of flip-flops, gates, and decoders. Control signals are produced sequentially as the unit transitions between states in response to instruction codes, timing signals, and external inputs. While it generates signals quickly, hardwired control units are difficult to modify or upgrade compared to microprogrammed units.
This document discusses subtractors in digital electronics. It defines a half subtractor and full subtractor, providing their truth tables, boolean expressions, and logic circuits. A half subtractor subtracts one single bit binary number from another and outputs the difference and borrow. A full subtractor subtracts three single bit binary numbers and outputs the difference and borrow. The document also describes how a parallel binary subtractor can be implemented by connecting multiple full subtractors to subtract binary numbers with multiple bits.
The document provides an introduction to PIC microcontrollers, including:
- The PIC16C6X/7X family uses a Harvard architecture with separate program and data memory buses, allowing fast instruction execution.
- The CPU contains registers like the Working Register, Status Register, FSR, and 8-level stack.
- Memory is organized into program memory, data memory (register files) and stack.
- Upon reset, the PIC initializes registers and jumps to address 0 to begin program execution. Resets ensure the PIC starts in a known state.
This document discusses memory reference instructions (MRI) and their implementation using microoperations. It defines MRI as instructions that operate on data stored in memory. Seven common MRI are described: AND to AC, ADD to AC, LDA, STA, BUN, BSA, and ISZ. Each MRI is broken down into its constituent microoperations, which are controlled by timing signals. The microoperations transfer data between memory, registers, and logic circuits. A control flow chart illustrates the sequencing of microoperations for each instruction type.
Simulation power analysis low power vlsiGargiKhanna1
The document discusses various simulation techniques used to estimate power dissipation at different levels of abstraction. It describes the tradeoff between computing resources and accuracy at different levels from algorithm to transistor level. SPICE circuit simulation provides the most accurate results but requires significant computing power. Higher levels of abstraction like gate level, switch level and architecture level analyses provide faster simulation speed at the cost of reduced accuracy. Power models are developed based on activities, component operations and data correlation to capture power at architecture level for large designs.
This document provides an overview of implementing a simplified MIPS processor with a memory-reference instructions, arithmetic-logical instructions, and control flow instructions. It discusses:
1. Using a program counter to fetch instructions from memory and reading register operands.
2. Executing most instructions via fetching, operand fetching, execution, and storing in a single cycle.
3. Building a datapath with functional units for instruction fetching, ALU operations, memory references, and branches/jumps.
4. Implementing control using a finite state machine that sets multiplexers and control lines based on the instruction.
Overview of Shift register and applicationsKarthik Kumar
Shift registers are used for temporary storage and can store multiple bits. There are two types: serial load and parallel load. Serial load registers shift data in or out one bit at a time on each clock cycle, while parallel load registers allow loading/output of all bits simultaneously. Common applications of shift registers include ring counters, Johnson counters, and pseudo-random pattern generators. A ring counter shifts data through the register and back to the first stage, counting through a fixed number of states. A Johnson counter also shifts data through the stages but with the last stage complement fed back to first, allowing it to count through more states than a ring counter.
This document provides a summary of digital logic design concepts including combinational logic, Boolean equations, Karnaugh maps, hazards, and NAND/NOR representation. It discusses combinational logic, Boolean equation forms, using Karnaugh maps to minimize logic functions, hazards that can occur in combinational circuits, and ways to remove hazards. It also explains that any logic function can be realized using only NAND or NOR gates and how basic NAND and NOR gates can be implemented using transistors.
This presentation discusses different types of microoperations that can be performed on data stored in registers. It describes arithmetic microoperations like addition, subtraction, and increment/decrement. Logic microoperations perform bit-wise operations on registers like selective set, clear, complement, and masking. Shift microoperations serially transfer data in a register left or right through logical, circular, and arithmetic shifts. Arithmetic shifts preserve a number's sign during multiplication and division by 2 during left and right shifts.
Mealy and Moore machines are types of finite state machines. A Mealy machine's output depends on its present state and input, while a Moore machine's output depends only on its present state. Mealy machines can be converted to Moore machines by breaking states with multiple outputs into multiple states, and vice versa by combining states with the same output. Both machine types have advantages and uses, with Mealy machines being faster but more expensive, and Moore machines being simpler but slower.
The document discusses some key concepts in VHDL, a hardware description language. It explains that in VHDL, an entity defines a module's external interface or ports, hiding its internal details. The architecture then describes the module's internal structure or behavior. There can be multiple architectures for a single entity. The architecture uses the ports from the entity and can contain additional internal signals and components.
The document discusses modules and ports in Verilog. It describes that a module defines distinct parts including module name, port list, port declarations, and optional parameters. Ports provide the interface for a module to communicate with its environment. There are two methods for connecting ports to external signals - by ordered list where signals must appear in the same order as ports, and by name where the order does not matter as long as port names match. Hierarchical names provide unique names for every identifier by denoting the design hierarchy with identifiers separated by periods.
This document discusses register transfer language (RTL) which provides a concise way to describe operations between registers in a computer using symbolic notation. It defines common registers like the memory address register (MAR) and program counter (PC). Information can be transferred between registers using arrows. Basic symbols are used to denote registers and parts of registers. Transfers can happen over a shared bus connecting all registers. Memory is represented as a device that is accessed using a memory address register to specify the location. RTL provides an organized way to describe the internal operations of a computer concisely and precisely.
DESIGN AND IMPLEMENTATION OF 64-BIT ARITHMETIC LOGIC UNIT ON FPGA USING VHDLsateeshkourav
The functions of fixed-point arithmetic were verified by
simulations with the single instruction test as the first
point. And then implemented fixed-point arithmetic with
FPGA. To handle more challenges nowadays and The
demand for complex tasks is increasing day by day to
increase the efficiency of a processor resulting in more
number of components manufactured on a single chip
according to Moore's law.
SPI is a serial bus standard established by Motorola and supported in silicon products from various manufacturers.
It is a synchronous serial data link that operates in full duplex (signals carrying data go in both directions simultaneously).
Devices communicate using a master/slave relationship, in which the master initiates the data frame. When the master generates a clock and selects a slave device, data may be transferred in either or both directions simultaneously.
1. The document discusses input and output streams in C++. It explains that streams act as an interface between a program and input/output devices, and that there are input streams to provide data to a program and output streams to send output from the program.
2. It then covers the hierarchy of stream classes in C++, including ios, istream, ostream, and iostream. It describes what each class contains and inherits from.
3. Unformatted I/O functions like get(), put(), and getline() are explained along with examples of how to use them for single character and line input/output.
Booth's algorithm is a method for multiplying two signed or unsigned integers in binary representation more efficiently than straightforward algorithms. It uses fewer additions and subtractions by representing the multiplicand as 2's complement numbers. The algorithm loads the multiplicand and multiplier into registers, initializes a third register to 0, and performs bitwise shifts and arithmetic operations (addition/subtraction of the multiplicand) on the registers based on the values of bits from the multiplier. This process builds up the product one bit at a time in a third register.
Cache memory is a small, fast memory located between the CPU and main memory. It stores copies of frequently used instructions and data to accelerate access and improve performance. There are different mapping techniques for cache including direct mapping, associative mapping, and set associative mapping. When the cache is full, replacement algorithms like LRU and FIFO are used to determine which content to remove. The cache can write to main memory using either a write-through or write-back policy.
Module 2 ARM CORTEX M3 Instruction Set and ProgrammingAmogha Bandrikalli
This document provides an overview of the ARM Cortex M3 instruction sets and assembly programming. It begins by explaining why assembly is commonly used, including the need for detailed control flow and efficiency. The outline presented covers assembly basics, instructions, useful instructions, and assembly and C language programming. It then discusses the various instruction sets of the Cortex M3 with examples. Specific instructions covered include data processing, branch, load/store, and 32-bit instructions. It also notes instructions not supported by the Cortex M3 such as coprocessor, state change, and hint instructions.
This document summarizes bivariate data and linear regression analysis. It introduces scatterplots and the Pearson correlation coefficient as ways to examine relationships between two variables. A positive correlation indicates that as one variable increases, so does the other, while a negative correlation means one variable increases as the other decreases. The least squares line provides the best fit linear relationship between two variables by minimizing the sum of squared residuals. Calculating the slope and y-intercept of this line allows predicting y-values from x-values. Examples using bus fare and distance data demonstrate these concepts.
This document provides an overview of regression analysis and two-way tables. It defines key concepts such as regression lines, correlation, residuals, and marginal and conditional distributions. Regression finds the linear relationship between two variables to make predictions. The least squares regression line minimizes the vertical distance between the data points and the line. Correlation and the coefficient of determination r2 measure how well the regression line fits the data. Two-way tables summarize the relationship between two categorical variables through marginal and conditional distributions.
The document contains a list of 23 microprocessor lab programs and 6 interfacing programs for an electronics and communication course. The programs cover topics like data transfer, arithmetic operations, sorting, prime number generation, string operations, matrix multiplication and more. The document provides contents, program descriptions and assembly language code for some of the programs.
A hardwired control unit generates control signals using finite state machines and a physically connected logic circuit. It consists of flip-flops, gates, and decoders. Control signals are produced sequentially as the unit transitions between states in response to instruction codes, timing signals, and external inputs. While it generates signals quickly, hardwired control units are difficult to modify or upgrade compared to microprogrammed units.
This document discusses subtractors in digital electronics. It defines a half subtractor and full subtractor, providing their truth tables, boolean expressions, and logic circuits. A half subtractor subtracts one single bit binary number from another and outputs the difference and borrow. A full subtractor subtracts three single bit binary numbers and outputs the difference and borrow. The document also describes how a parallel binary subtractor can be implemented by connecting multiple full subtractors to subtract binary numbers with multiple bits.
The document provides an introduction to PIC microcontrollers, including:
- The PIC16C6X/7X family uses a Harvard architecture with separate program and data memory buses, allowing fast instruction execution.
- The CPU contains registers like the Working Register, Status Register, FSR, and 8-level stack.
- Memory is organized into program memory, data memory (register files) and stack.
- Upon reset, the PIC initializes registers and jumps to address 0 to begin program execution. Resets ensure the PIC starts in a known state.
This document discusses memory reference instructions (MRI) and their implementation using microoperations. It defines MRI as instructions that operate on data stored in memory. Seven common MRI are described: AND to AC, ADD to AC, LDA, STA, BUN, BSA, and ISZ. Each MRI is broken down into its constituent microoperations, which are controlled by timing signals. The microoperations transfer data between memory, registers, and logic circuits. A control flow chart illustrates the sequencing of microoperations for each instruction type.
Simulation power analysis low power vlsiGargiKhanna1
The document discusses various simulation techniques used to estimate power dissipation at different levels of abstraction. It describes the tradeoff between computing resources and accuracy at different levels from algorithm to transistor level. SPICE circuit simulation provides the most accurate results but requires significant computing power. Higher levels of abstraction like gate level, switch level and architecture level analyses provide faster simulation speed at the cost of reduced accuracy. Power models are developed based on activities, component operations and data correlation to capture power at architecture level for large designs.
This document provides an overview of implementing a simplified MIPS processor with a memory-reference instructions, arithmetic-logical instructions, and control flow instructions. It discusses:
1. Using a program counter to fetch instructions from memory and reading register operands.
2. Executing most instructions via fetching, operand fetching, execution, and storing in a single cycle.
3. Building a datapath with functional units for instruction fetching, ALU operations, memory references, and branches/jumps.
4. Implementing control using a finite state machine that sets multiplexers and control lines based on the instruction.
Overview of Shift register and applicationsKarthik Kumar
Shift registers are used for temporary storage and can store multiple bits. There are two types: serial load and parallel load. Serial load registers shift data in or out one bit at a time on each clock cycle, while parallel load registers allow loading/output of all bits simultaneously. Common applications of shift registers include ring counters, Johnson counters, and pseudo-random pattern generators. A ring counter shifts data through the register and back to the first stage, counting through a fixed number of states. A Johnson counter also shifts data through the stages but with the last stage complement fed back to first, allowing it to count through more states than a ring counter.
This document provides a summary of digital logic design concepts including combinational logic, Boolean equations, Karnaugh maps, hazards, and NAND/NOR representation. It discusses combinational logic, Boolean equation forms, using Karnaugh maps to minimize logic functions, hazards that can occur in combinational circuits, and ways to remove hazards. It also explains that any logic function can be realized using only NAND or NOR gates and how basic NAND and NOR gates can be implemented using transistors.
This presentation discusses different types of microoperations that can be performed on data stored in registers. It describes arithmetic microoperations like addition, subtraction, and increment/decrement. Logic microoperations perform bit-wise operations on registers like selective set, clear, complement, and masking. Shift microoperations serially transfer data in a register left or right through logical, circular, and arithmetic shifts. Arithmetic shifts preserve a number's sign during multiplication and division by 2 during left and right shifts.
Mealy and Moore machines are types of finite state machines. A Mealy machine's output depends on its present state and input, while a Moore machine's output depends only on its present state. Mealy machines can be converted to Moore machines by breaking states with multiple outputs into multiple states, and vice versa by combining states with the same output. Both machine types have advantages and uses, with Mealy machines being faster but more expensive, and Moore machines being simpler but slower.
The document discusses some key concepts in VHDL, a hardware description language. It explains that in VHDL, an entity defines a module's external interface or ports, hiding its internal details. The architecture then describes the module's internal structure or behavior. There can be multiple architectures for a single entity. The architecture uses the ports from the entity and can contain additional internal signals and components.
The document discusses modules and ports in Verilog. It describes that a module defines distinct parts including module name, port list, port declarations, and optional parameters. Ports provide the interface for a module to communicate with its environment. There are two methods for connecting ports to external signals - by ordered list where signals must appear in the same order as ports, and by name where the order does not matter as long as port names match. Hierarchical names provide unique names for every identifier by denoting the design hierarchy with identifiers separated by periods.
This document discusses register transfer language (RTL) which provides a concise way to describe operations between registers in a computer using symbolic notation. It defines common registers like the memory address register (MAR) and program counter (PC). Information can be transferred between registers using arrows. Basic symbols are used to denote registers and parts of registers. Transfers can happen over a shared bus connecting all registers. Memory is represented as a device that is accessed using a memory address register to specify the location. RTL provides an organized way to describe the internal operations of a computer concisely and precisely.
DESIGN AND IMPLEMENTATION OF 64-BIT ARITHMETIC LOGIC UNIT ON FPGA USING VHDLsateeshkourav
The functions of fixed-point arithmetic were verified by
simulations with the single instruction test as the first
point. And then implemented fixed-point arithmetic with
FPGA. To handle more challenges nowadays and The
demand for complex tasks is increasing day by day to
increase the efficiency of a processor resulting in more
number of components manufactured on a single chip
according to Moore's law.
SPI is a serial bus standard established by Motorola and supported in silicon products from various manufacturers.
It is a synchronous serial data link that operates in full duplex (signals carrying data go in both directions simultaneously).
Devices communicate using a master/slave relationship, in which the master initiates the data frame. When the master generates a clock and selects a slave device, data may be transferred in either or both directions simultaneously.
1. The document discusses input and output streams in C++. It explains that streams act as an interface between a program and input/output devices, and that there are input streams to provide data to a program and output streams to send output from the program.
2. It then covers the hierarchy of stream classes in C++, including ios, istream, ostream, and iostream. It describes what each class contains and inherits from.
3. Unformatted I/O functions like get(), put(), and getline() are explained along with examples of how to use them for single character and line input/output.
Booth's algorithm is a method for multiplying two signed or unsigned integers in binary representation more efficiently than straightforward algorithms. It uses fewer additions and subtractions by representing the multiplicand as 2's complement numbers. The algorithm loads the multiplicand and multiplier into registers, initializes a third register to 0, and performs bitwise shifts and arithmetic operations (addition/subtraction of the multiplicand) on the registers based on the values of bits from the multiplier. This process builds up the product one bit at a time in a third register.
Cache memory is a small, fast memory located between the CPU and main memory. It stores copies of frequently used instructions and data to accelerate access and improve performance. There are different mapping techniques for cache including direct mapping, associative mapping, and set associative mapping. When the cache is full, replacement algorithms like LRU and FIFO are used to determine which content to remove. The cache can write to main memory using either a write-through or write-back policy.
Module 2 ARM CORTEX M3 Instruction Set and ProgrammingAmogha Bandrikalli
This document provides an overview of the ARM Cortex M3 instruction sets and assembly programming. It begins by explaining why assembly is commonly used, including the need for detailed control flow and efficiency. The outline presented covers assembly basics, instructions, useful instructions, and assembly and C language programming. It then discusses the various instruction sets of the Cortex M3 with examples. Specific instructions covered include data processing, branch, load/store, and 32-bit instructions. It also notes instructions not supported by the Cortex M3 such as coprocessor, state change, and hint instructions.
This document summarizes bivariate data and linear regression analysis. It introduces scatterplots and the Pearson correlation coefficient as ways to examine relationships between two variables. A positive correlation indicates that as one variable increases, so does the other, while a negative correlation means one variable increases as the other decreases. The least squares line provides the best fit linear relationship between two variables by minimizing the sum of squared residuals. Calculating the slope and y-intercept of this line allows predicting y-values from x-values. Examples using bus fare and distance data demonstrate these concepts.
This document provides an overview of regression analysis and two-way tables. It defines key concepts such as regression lines, correlation, residuals, and marginal and conditional distributions. Regression finds the linear relationship between two variables to make predictions. The least squares regression line minimizes the vertical distance between the data points and the line. Correlation and the coefficient of determination r2 measure how well the regression line fits the data. Two-way tables summarize the relationship between two categorical variables through marginal and conditional distributions.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
Correlation & Regression for Statistics Social Sciencessuser71ac73
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
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Chapter 10: Correlation and Regression
10.2: Regression
This document defines and explains various types of regression analysis including linear, logistic, polynomial, stepwise, ridge and lasso regression. It discusses the key differences between correlation and regression. It also covers topics such as the least squares method, R-squared/coefficient of determination, adjusted R-squared, limitations of regression analysis and applications of regression analysis.
This document discusses correlation and regression analysis. It defines correlation as assessing the relationship between two variables, while regression determines how well one variable can predict another. Correlation does not imply causation. Pearson's r standardizes the covariance between variables and ranges from -1 to 1, indicating the strength and direction of their linear relationship. Regression finds the best-fitting linear relationship through the least squares method to minimize residuals and predict one variable from another. It provides the slope and intercept of the regression line. The coefficient of determination, r-squared, indicates how well the regression model fits the data.
This document discusses summarizing bivariate data using scatterplots and correlation. It provides an example of fare data from a bus company that is modeled using linear and nonlinear regression. Linear regression finds a strong positive correlation between distance and fare, but the relationship is better modeled nonlinearly using the logarithm of distance. The nonlinear model accounts for 96.9% of variation in fares compared to 84.9% for the linear model.
This document summarizes key concepts from Chapter 5 of Jamri AB on correlation and simple linear regression. It introduces correlation as a measure of the strength of the linear relationship between two variables. It discusses scatter diagrams, the coefficient of correlation (r), and Pearson's product-moment correlation coefficient and Spearman's rank correlation coefficient as methods to calculate r. It also covers the coefficient of determination (r^2), linear regression analysis to predict relationships, and calculating the regression equation coefficients a and b. Examples are provided to demonstrate calculating r and the regression equation from sets of data.
To get a copy of the slides for free Email me at: japhethmuthama@gmail.com
You can also support my PhD studies by donating a 1 dollar to my PayPal.
PayPal ID is japhethmuthama@gmail.com
To get a copy of the slides for free Email me at: japhethmuthama@gmail.com
You can also support my PhD studies by donating a 1 dollar to my PayPal.
PayPal ID is japhethmuthama@gmail.com
Simple Regression presentation is a
partial fulfillment to the requirement in PA 297 Research for Public Administrators, presented by Atty. Gayam , Dr. Cabling and Mr. Cagampang
This document provides an overview of regression analysis. It defines regression analysis as a predictive modeling technique used to investigate relationships between dependent and independent variables. It describes simple linear regression as involving one independent variable and one dependent variable, with the goal of finding the best fitting straight line through the data points. An example is provided to demonstrate how to conduct a simple linear regression to predict population in the year 2005 based on population data from previous years.
The document provides an introduction to unsupervised learning and reinforcement learning. It then discusses eigen values and eigen vectors, showing how to calculate them from a matrix. It provides examples of covariance matrices and using Gaussian elimination to solve for eigen vectors. Finally, it discusses principal component analysis and different clustering algorithms like K-means clustering.
Cross validation is a technique for evaluating machine learning models by splitting the dataset into training and validation sets and training the model multiple times on different splits, to reduce variance. K-fold cross validation splits the data into k equally sized folds, where each fold is used once for validation while the remaining k-1 folds are used for training. Leave-one-out cross validation uses a single observation from the dataset as the validation set. Stratified k-fold cross validation ensures each fold has the same class proportions as the full dataset. Grid search evaluates all combinations of hyperparameters specified as a grid, while randomized search samples hyperparameters randomly within specified ranges. Learning curves show training and validation performance as a function of training set size and can diagnose underfitting
This document provides an overview of supervised machine learning algorithms for classification, including logistic regression, k-nearest neighbors (KNN), support vector machines (SVM), and decision trees. It discusses key concepts like evaluation metrics, performance measures, and use cases. For logistic regression, it covers the mathematics behind maximum likelihood estimation and gradient descent. For KNN, it explains the algorithm and discusses distance metrics and a numerical example. For SVM, it outlines the concept of finding the optimal hyperplane that maximizes the margin between classes.
The document provides information on solving the sum of subsets problem using backtracking. It discusses two formulations - one where solutions are represented by tuples indicating which numbers are included, and another where each position indicates if the corresponding number is included or not. It shows the state space tree that represents all possible solutions for each formulation. The tree is traversed depth-first to find all solutions where the sum of the included numbers equals the target sum. Pruning techniques are used to avoid exploring non-promising paths.
The document discusses the greedy method and its applications. It begins by defining the greedy approach for optimization problems, noting that greedy algorithms make locally optimal choices at each step in hopes of finding a global optimum. Some applications of the greedy method include the knapsack problem, minimum spanning trees using Kruskal's and Prim's algorithms, job sequencing with deadlines, and finding the shortest path using Dijkstra's algorithm. The document then focuses on explaining the fractional knapsack problem and providing a step-by-step example of solving it using a greedy approach. It also provides examples and explanations of Kruskal's algorithm for finding minimum spanning trees.
The document describes various divide and conquer algorithms including binary search, merge sort, quicksort, and finding maximum and minimum elements. It begins by explaining the general divide and conquer approach of dividing a problem into smaller subproblems, solving the subproblems independently, and combining the solutions. Several examples are then provided with pseudocode and analysis of their divide and conquer implementations. Key algorithms covered in the document include binary search (log n time), merge sort (n log n time), and quicksort (n log n time on average).
What is an Algorithm
Time Complexity
Space Complexity
Asymptotic Notations
Recursive Analysis
Selection Sort
Insertion Sort
Recurrences
Substitution Method
Master Tree Method
Recursion Tree Method
This document provides an outline for a machine learning syllabus. It includes 14 modules covering topics like machine learning terminology, supervised and unsupervised learning algorithms, optimization techniques, and projects. It lists software and hardware requirements for the course. It also discusses machine learning applications, issues, and the steps to build a machine learning model.
The document discusses problem-solving agents and their approach to solving problems. Problem-solving agents (1) formulate a goal based on the current situation, (2) formulate the problem by defining relevant states and actions, and (3) search for a solution by exploring sequences of actions that lead to the goal state. Several examples of problems are provided, including the 8-puzzle, robotic assembly, the 8 queens problem, and the missionaries and cannibals problem. For each problem, the relevant states, actions, goal tests, and path costs are defined.
The simplex method is a linear programming algorithm that can solve problems with more than two decision variables. It works by generating a series of solutions, called tableaus, where each tableau corresponds to a corner point of the feasible solution space. The algorithm starts at the initial tableau, which corresponds to the origin. It then shifts to adjacent corner points, moving in the direction that optimizes the objective function. This process of generating new tableaus continues until an optimal solution is found.
The document discusses functions and the pigeonhole principle. It defines what a function is, how functions can be represented graphically and with tables and ordered pairs. It covers one-to-one, onto, and bijective functions. It also discusses function composition, inverse functions, and the identity function. The pigeonhole principle states that if n objects are put into m containers where n > m, then at least one container must hold more than one object. Examples are given to illustrate how to apply the principle to problems involving months, socks, and selecting numbers.
The document discusses relations and their representations. It defines a binary relation as a subset of A×B where A and B are nonempty sets. Relations can be represented using arrow diagrams, directed graphs, and zero-one matrices. A directed graph represents the elements of A as vertices and draws an edge from vertex a to b if aRb. The zero-one matrix representation assigns 1 to the entry in row a and column b if (a,b) is in the relation, and 0 otherwise. The document also discusses indegrees, outdegrees, composite relations, and properties of relations like reflexivity.
This document discusses logic and propositional logic. It covers the following topics:
- The history and applications of logic.
- Different types of statements and their grammar.
- Propositional logic including symbols, connectives, truth tables, and semantics.
- Quantifiers, universal and existential quantification, and properties of quantifiers.
- Normal forms such as disjunctive normal form and conjunctive normal form.
- Inference rules and the principle of mathematical induction, illustrated with examples.
1. Set theory is an important mathematical concept and tool that is used in many areas including programming, real-world applications, and computer science problems.
2. The document introduces some basic concepts of set theory including sets, members, operations on sets like union and intersection, and relationships between sets like subsets and complements.
3. Infinite sets are discussed as well as different types of infinite sets including countably infinite and uncountably infinite sets. Special sets like the empty set and power sets are also covered.
The document discusses uncertainty and probabilistic reasoning. It describes sources of uncertainty like partial information, unreliable information, and conflicting information from multiple sources. It then discusses representing and reasoning with uncertainty using techniques like default logic, rules with probabilities, and probability theory. The key approaches covered are conditional probability, independence, conditional independence, and using Bayes' rule to update probabilities based on new evidence.
The document outlines the objectives, outcomes, and learning outcomes of a course on artificial intelligence. The objectives include conceptualizing ideas and techniques for intelligent systems, understanding mechanisms of intelligent thought and action, and understanding advanced representation and search techniques. Outcomes include developing an understanding of AI building blocks, choosing appropriate problem solving methods, analyzing strengths and weaknesses of AI approaches, and designing models for reasoning with uncertainty. Learning outcomes include knowledge, intellectual skills, practical skills, and transferable skills in artificial intelligence.
Planning involves representing an initial state, possible actions, and a goal state. A planning agent uses a knowledge base to select action sequences that transform the initial state into a goal state. STRIPS is a common planning representation that uses predicates to describe states and logical operators to represent actions and their effects. A STRIPS planning problem specifies the initial state, goal conditions, and set of operators. A solution is a sequence of ground operator instances that produces the goal state from the initial state.
This presentation is about health care analysis using sentiment analysis .
*this is very useful to students who are doing project on sentiment analysis
*
Optimizing Feldera: Integrating Advanced UDFs and Enhanced SQL Functionality ...mparmparousiskostas
This report explores our contributions to the Feldera Continuous Analytics Platform, aimed at enhancing its real-time data processing capabilities. Our primary advancements include the integration of advanced User-Defined Functions (UDFs) and the enhancement of SQL functionality. Specifically, we introduced Rust-based UDFs for high-performance data transformations and extended SQL to support inline table queries and aggregate functions within INSERT INTO statements. These developments significantly improve Feldera’s ability to handle complex data manipulations and transformations, making it a more versatile and powerful tool for real-time analytics. Through these enhancements, Feldera is now better equipped to support sophisticated continuous data processing needs, enabling users to execute complex analytics with greater efficiency and flexibility.
Essential Skills for Family Assessment - Marital and Family Therapy and Couns...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
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ML Module 3.pdf
1. MODULE 3
Supervised ML – Regression
Shiwani Gupta
Use case
Simple Linear
Gradient Descent
Evaluation Metric
Multi Linear, Polynomial
Regularization
2. USE CASES
A hospital may be interested in finding how total cost of a patient varies with severity of disease.
Insurance companies would like to understand association between healthcare cost and ageing.
An organization may be interested in finding relationship between revenue generated from a product and
features such as price, promotional amount spent, competitor’s price of similar product, etc.
Restaurants would like to know relationship between customer waiting time after placing the order and the
revenue generated.
E-commerce companies like: Amazon, BigBasket, Flipkart , etc. would like to understand the relationship
between revenue generated and features like no. of customer visit to portal, no. of clicks on products, no.
of items on sale, av. discount percentage etc.
Bank and other financial institutions would like to understand the impact of variables such as
unemployment rate, marital status, bank balance, etc. on percentage of Non Performing Assets, etc.
2
4. LINEAR REGRESSION
Linear Regression is a Supervised Machine Learning algorithm for
predictive modelling.
It tries to find out the best linear relationship that describes the data you have
(Scatter Plot).
It assumes that there exists a linear relationship between a dependent
variable (usually called y) and independent variable(s) (usually called X).
The value of the dependent / response / outcome variable of a Linear
Regression model is a continuous value / quantitative in nature i.e. real
numbers.
Linear Regression model represents linear relationship between a dependent
variable and independent / predictor / explanatory variable(s) via a sloped
straight line.
The sloped straight line representing the linear relationship that fits the given
data best is called a Regression Line / Best Fit Line.
Based on the number of independent variables, there are two types of Linear
Regression
4
5. INFERENCE ABOUT THE REGRESSION MODEL
When a scatter plot shows a linear relationship between a quantitative explanatory variable x and a
quantitative response variable y, we can use the least square line fitted to the data to predict y for a given
value of x.
We think of the least square line we calculated from the sample as an estimate of a regression line for the
population.
Just as the sample mean is an estimate of the population mean µ.
We will write the population regression line as
The numbers and are parameters that describe the population.
We will write the least-squares line fitted to sample data as
This notation reminds us that the intercept b0 of the fitted line estimates the intercept 0 of the
population line, and the slope b1 estimates the slope 1 respectively.
x
x
1
0
0
1
x
b
b 1
0
5
6. SIMPLE LINEAR REGRESSION
A statistical method to summarize and study the functional relationship b/w 2 cont. variables.
May be linear or nonlinear (eg. Population growth over time)
The dependent variable depends only on a single independent variable.
The form of the model is: y = β0 + β1X eg. V=I*R, Circumference = 2*pi*r, C=(F-32)*5/9, etc.
y is a dependent variable.
X is an independent variable.
β0 and β1 are the regression coefficients.
β0 is the intercept or the bias that fixes the offset to a line. It is the av. y value for Xmean = 0
β1 is the slope or weight that specifies the factor by which X has an impact on y.
The values of the regression parameters 0, and 1 are not known.
We estimate them from data.
6
8. REGRESSION LINE
We will write an estimated regression line based on sample data as
Least Squares Method give us the “best” estimated line for our set of sample data.
The method of least squares chooses the values for b0 and b1 to minimize the Sum of Squared Errors
Using calculus, we obtain the estimating formulas:
x
b
b
y 1
0
ˆ
2
1
1
0
1
2
)
ˆ
(
n
i
n
i
i
i x
b
b
y
y
y
SSE
n
i
n
i
i
i
n
i
n
i
n
i
i
i
i
i
n
i
i
n
i
i
i
x
x
n
y
x
y
x
n
x
x
y
y
x
x
b
1 1
2
2
1 1 1
1
2
1
1
)
(
)
(
)
)(
(
x
b
y
b 1
0
Fitted regression line can be used to estimate y for a given value of x.
𝑀𝑆𝐸 = (1/𝑛)
𝑖=1
𝑛
(𝑦𝑖 − 𝑦𝑖)2 𝑀𝐴𝐸 = (1/𝑛)
𝑖=1
𝑛
|𝑦𝑖 − 𝑦 |
8
9. STEPS TO ESTABLISH A LINEAR RELATION
Gather sample of observed height and corresponding weight.
Create relationship model.
Find coefficients from model and establish mathematical equation.
Get summary of model to compute av. prediction error … Residual.
Predict weight
height weight
151 63
174 81
138 56
186 91
128 47
136 57
179 76
163 72
152 62
131 48
2
1
1
0
n
i
x
b
b
y
Q
We want to penalize the points which are
farther from the regression line much more
than the points which lie close to the line.
9
11. GRADIENT DESCENT
used to minimize the cost function Q
STEPS:
1. Random initialization for θ1 and θ0.
2. Measure how the cost function changes with change in it’s parameters by computing the partial derivatives of
cost function w.r.t to the parameters θ₀, θ₁, … , θₙ.
3. After computing derivative, update parameters θj: = θj−α ∂/∂θj Q(θ0,θ1) for j=0,1 where α, learning rate, a
positive no. and a step to update parameters.
4. Repeat process of Simultaneous update of θ1 and θ0, until convergence.
α too small, too much time, α too large, failure to converge.
11
12. GRADIENT DESCENT FOR UNIVARIATE LINEAR REGRESSION
Hypothesis hθ(x)=θ0+θ1x
Cost Function J(θ0,θ1)=(1/2*m) * ∑i=1
m(hθ(x(i))−y(i))2
Gradient Descent to minimize cost function for Linear Regression model
Compute derivative for j=0, j=1
12
13. Dispersion of observed variable around mean How well our line fits data
total variability of the data is equal to the variability
explained by the regression line plus the unexplained
variability, known as error.
MODEL
EVALUATION
13
14. COEFFICIENT OF DETERMINATION
Recall that SST measures the total variations in yi when no account of the independent
variable x is taken.
SSE measures the variation in the yi when a regression model with the independent variable x
is used.
A natural measure of the effect of x in reducing the variation in y can be defined as:
R2 is called the coefficient of determination / goodness of fit.
0 SSE SST, it follows that:
We may interpret R2 as the proportionate reduction of total variability in y associated with the
use of the independent variable x.
The larger is R2, the more is the total variation of y reduced by including the variable x in the
model.
SST
SSE
SST
SSR
SST
SSE
SST
R
1
2
𝟎 ≤ 𝑅2
≤ 𝟏
14
15. COEFFICIENT OF DETERMINATION
If all the observations fall on the fitted regression line, SSE = 0 and R2 = 1.
If the slope of the fitted regression line
b1 = 0 so that , SSE=SST and R2 = 0.
The closer R2 is to 1, the greater is said to be the degree of linear association
between x and y.
The square root of R2 is called the coefficient of correlation (r).
y
yi
ˆ
2
R
r
2
2
2
2
2
2
)
(
)
(
)
(
)
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)(
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y
y
n
x
x
n
y
x
xy
n
r
y
y
x
x
y
y
x
x
r
15
16. MODEL EVALUATION : R-SQUARED
height (cm) weight (kg) ypredicted SSE = (y-ypred)2 SST = (y-ymean)2 SSR = (ypred-ymean)2
151 63 63.4111 0.16901143 5.29 3.56790543
174 81 78.9271 4.29674858 246.49 185.698945
138 56 54.6412 1.84639179 86.49 113.610444
186 91 87.0225 15.8208245 660.49 471.865268
128 47 47.8951 0.80116821 334.89 302.93124
136 57 53.292 13.7495606 68.89 144.193025
179 76 82.3002 39.692394 114.49 289.00646
163 72 71.5064 0.24361134 44.89 38.5197733
152 62 64.0857 4.35022792 10.89 1.47447592
131 48 49.9189 3.68221559 299.29 236.57793
ymean sum sum sum
65.3 82.652154 1872.1 1787.44547
R2 = SSR/SST = 0.95478
measures the
proportion of the
variation in your
dependent
variable
explained by all
your independent
variables in the
model
R2 = [0,1]
16
17. ESTIMATION OF MEAN RESPONSE
The weekly advertising expenditure (x) and weekly sales (y) are presented in the following table:
From the table, the least squares estimates of the regression coefficients are:
y x
1250 41
1380 54
1425 63
1425 54
1450 48
1300 46
1400 62
1510 61
1575 64
1650 71
818755
14365
32604
564
10 2
xy
y
x
x
n
8
.
10
)
564
(
)
32604
(
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)(
564
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818755
(
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y
x
xy
n
b 828
)
4
.
56
(
8
.
10
5
.
1436
0
b
The estimated regression function is:
This means that if weekly advertising expenditure is increased by $1 we would expect the weekly sales to increase by $10.8.
Fitted values for the sample data are obtained by substituting the x value into the estimated regression function.
For example, if the advertising expenditure is $50, then the estimated Sales is:
This is called the point estimate (forecast) of the mean response (sales).
e
Expenditur
8
.
10
828
Sales
10.8x
828
ŷ
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(
8
.
10
828
Sales
x
b
y
b 1
0
17
18. EXAMPLE: SOLVE
• The primary goal of Quantitative Analysis is to use current information about a
phenomenon to predict its future behavior.
• Current information is usually in the form of data.
• In a simple case, when the data forms a set of pairs of numbers, we may interpret
them as representing the observed values of an independent (or predictor) variable
X and a dependent (or response) variable y.
• The goal of the analyst who studies the data is to find a functional relation
between the response variable y and the predictor variable x.
lot size Man-hours
30 73
20 50
60 128
80 170
40 87
50 108
60 135
30 69
70 148
60 132
0
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80 90
Man-Hour
Lot size
Statistical relation between Lot size and Man-Hour
)
(x
f
y
18
19. EXAMPLE: RETAIL SALES AND FLOOR SPACE
It is customary in retail operations to assess the performance of stores partly in terms of their annual
sales relative to their floor area (square feet).
We might expect sales to increase linearly as stores get larger, with of course individual variation among
the stores of same size.
The regression model for a population of stores says that SALES = 0 + 1 AREA +
The slope 1 is rate of change: it is the expected increase in annual sales associated with each additional
square foot of floor space.
The intercept 0 is needed to describe the line but has no statistical importance because no stores have
area close to zero.
Floor space does not completely determine sales. The term in the model accounts for difference among
individual stores with the same floor space. A store’s location, for example, is important.
Residual: The difference between the observed value yi and the corresponding fitted value
Residuals are highly useful for studying whether a given regression model is appropriate for the
data at hand. i
i
i y
y
e ˆ
i
ŷ
19
20. ANALYSIS OF RESIDUAL
To examine whether the regression model is appropriate for the data being analyzed, we can check residual plots.
Residual plots are:
A scatterplot of the residuals
Plot residuals against the fitted values.
Plot residuals against the independent variable.
Plot residuals over time if the data are chronological.
The residuals should have no systematic pattern. Eg. The residual plot below shows a scatter of the points with no
individual observations or systematic change as x increases.
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 20 40 60
Residuals
Degree Days
Degree Days Residual Plot
20
21. RESIDUAL PLOTS
The points in this residual plot have a curve pattern, so a straight line fits
poorly
The points in this plot show more spread for larger values of the
explanatory variable x, so prediction will be less accurate when x is
large.
21
22. EXAMPLE: DO WAGES RISE WITH EXPERIENCE?
Many factors affect the wages of
workers: the industry they work in,
their type of job, their education,
their experience, and changes in
general levels of wages. We will
look at a sample of 59 married
women who hold customer service
jobs in Indiana banks. The table
gives their weekly wages at a
specific point in time also their
Length Of Service with their
employer, in months. The size of
the place of work is recorded
simply as “large” (100 or more
workers) or “small.” Because
industry, job type, and the time of
measurement are the same for all 59
subjects, we expect to see a clear
relationship between wages and
length of service.
22
23. EXAMPLE: DO WAGES RISE WITH EXPERIENCE?
From previous table we have:
The least squares estimates of the regression coefficients are:
1719376
9460467
23069
451031
4159
59
2
2
xy
y
y
x
x
n
x
b
y
b0
2
2
1
)
( x
x
n
y
x
xy
n
b
2
)
ˆ
( i
i y
y
SSE
2
)
( y
y
SST i
2
)
ˆ
( y
y
SSR i
23
24. USING THE REGRESSION LINE
One of the most common reasons to fit a line to data is to predict the response to a particular value of
the explanatory variable.
In our example, the least square line for predicting the weekly earnings for female bank customer
service workers from their length of service is
For a length of service of 125 months, our least-squares regression equation gives
x
y 5905
.
0
4
.
349
ˆ
per week
423
$
)
125
)(
5905
(.
4
.
349
ˆ
y
The measure of variation in the data around the fitted regression line
SSE = 36124.76 SST = 128552.5
If SST = 0, all observations are the same (No variability).
The greater is SST, the greater is the variation among the y values.
74
.
92427
76
.
36124
5
.
128552
SSE
SST
SSR
SSR is the variation among predicted responses. The predicted responses lie on the least-square
line. They show how y moves in response to x.
The larger is SSR relative to SST, the greater is the role of regression line in explaining the total
variability in y observations.
This indicates that most of variability in weekly sales can be explained by the relation between
the weekly advertising expenditure and the weekly sales.
R2 = SSR/SST = .719
24
25. MULTIPLE LINEAR REGRESSION
The dependent variable depends on more than one independent variables.
The form of the model is: y = b0 + b1x1 + b2x2 + b3x3 + …… + bnxn
May be Linear or nonlinear.
Here,
y is a dependent variable.
x1, x2, …., xn are independent variables.
b0, b1,…, bn are the regression coefficients.
bj (1<=j<=n) is the slope or weight that specifies the factor by which Xj has an impact on Y.
25
26. POLYNOMIAL REGRESSION
y= b0+b1x1+ b2x1
2+ b2x1
3+...... bnx1
n
Special case of Multiple Linear Regression.
We add some polynomial terms to the Multiple Linear regression equation to convert it into Polynomial Regression.
Linear model with some modification in order to increase the accuracy.
Training data is of non-linear nature.
In Polynomial regression, the original features are converted into Polynomial features of required degree (2,3,..,n) and
then modeled using a Linear model.
If we apply a linear model on a linear dataset, then it provides us good result, but if we apply the same model without
any modification on a non-linear dataset, then it will produce a drastic output. Due to which loss function will increase,
the error rate will be high, and the accuracy will decrease.
A Polynomial Regression algorithm is also called Polynomial Linear Regression because it does not depend on the
variables, instead, it depends on the coefficients, which are arranged in a linear fashion.
26
27. REGULARIZATION
To avoid overfitting of training data and hence enhance generalization performance.
Since model tries to capture noise, that doesn’t represent true properties of data.
Regularization is a form of regression that constraints/ regularizes/ shrinks coefficient estimates towards zero.
Y represents the learned relation, β represents the coefficient estimates for
different variables or predictors (X).
Coefficients are chosen so as to minimize loss function
27
28. • Eg. a person’s height and weight, age and sales price of a car, or years of education
and annual income
• Doesn’t affect DT
• kNN affected
• Cause
• Insufficient data
• Dummy variables
• Including a variable in the regression that is actually a combination of two
other variables.
• Identify (corr > 0.4, Variance Inflation Factor score > 5 high correlation )
• Sol
• Feature selection
• PCA
• More data
• Ridge regression reduces magnitude of model coefficients 28
29. RIDGE REGULARIZATION (L2 NORM)
Used when data suffers from multicollinearity
RSS is modified by adding shrinkage quantity, λ (tuning parameter) that decides how much we want to penalize the
flexibility of our model, intercept β0, is a measure of the mean value of the response when xi1 = xi2 = …= xip = 0.
If we want to minimize the above function, these coefficients need to be small.
When λ = 0, the penalty term has no effect, and the estimates produced by ridge regression will be equal to least squares.
However, as λ→∞, the impact of the shrinkage penalty grows, and ridge regression coefficient estimates will approach
zero.
Note: we need to standardize the predictors or bring the predictors to the same scale before performing ridge regression.
Disadvantage: model interpretability
29
30. LASSO REGULARIZATION (L1 NORM)
Least Absolute Shrinkage and Selection Operator
This variation differs from ridge regression only in penalizing the high coefficients.
It uses |βj| (modulus) instead of squares of β, as its penalty.
Lasso method also performs variable selection and is said to yield sparse models.
30
31. RIDGE LASSO COMPARISON
Ridge Regression can be thought of as solving an equation, where summation of squares of coefficients is less than or equal to
s. And Lasso can be thought of as an equation where summation of modulus of coefficients is less than or equal to s. Here, s is a
constant that exists for each value of shrinkage factor λ. These equations are also referred to as constraint functions.
Consider there are 2 parameters in a given problem. Then according to above formulation:
Ridge regression is expressed by β1² + β2² ≤ s. This implies that ridge regression coefficients have the smallest RSS (loss
function) for all points that lie within the circle given by β1² + β2² ≤ s.
for Lasso, the equation becomes, |β1|+|β2| ≤ s. This implies that lasso coefficients have the smallest RSS (loss function) for all
points that lie within the diamond given by |β1|+|β2|≤ s.
Image shows the constraint functions (green areas), for Lasso (left) and Ridge regression (right), along with contours for RSS
(red ellipse). The black point denotes that the least square error is minimized at that point and as we can see that it increases
quadratically as we move away from it and the regularization term is minimized at the origin where all the parameters are
zero
Since Ridge Regression has a circular constraint with no sharp points,
this intersection will not generally occur on an axis, and so ridge
regression coefficient estimates will be exclusively non-zero.
However, Lasso constraint has corners at each of the axes, and so the
ellipse will often intersect the constraint region at an axis. When this
occurs, one of the coefficients will equal zero.
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32. BENEFIT
Regularization significantly reduces the variance of model, without substantial increase in its bias.
The tuning parameter λ, controls the impact on bias and variance.
As the value of λ rises, it reduces the value of coefficients and thus reducing the variance.
Till a point, this increase in λ is beneficial as it is only reducing the variance (hence avoiding
overfitting), without loosing any important properties in the data.
But after certain value, the model starts loosing important properties, giving rise to bias in the model and
thus underfitting. Therefore, the value of λ should be carefully selected.
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33. SUMMATIVE ASSESSMENT
3 Consider the following dataset showing relationship
between food intake (lb) of cows and milk yield (lb).
Estimate the parameters for the linear regression model
for the dataset:
Food (lb) Milk Yield (lb)
4 3.0
6 5.5
10 6.5
12 9.0
4 Fit a Linear Regression model for following relation
between mother’s Estirol level and birth weight of child
for following data:
Estirol (mg/24 hr) Birth weight (g/100)
1 1
2 1
3 2
4 2
5 4
5 Create a relationship model for given data to find
relationship b/w height and weight of students. Compute
Karl Pearson coefficient and Coefficient of determination.
REFER SLIDE 9
6 State benefits of regularization for avoiding overfitting
in Linear Regression. State mathematical formulation of
Regularization.
7 Explain steps of Gradient Descent Algorithm.
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1.The rent of a property is related to its area. Given the area in square feet and rent in dollars, find the
relationship between area and rent using the concept of linear regression. Also predict the rent for a
property of 790 ft square.
2. The marks obtained by a student is dependent on his/her study time. Given the study time in minutes
and marks out of 2000. Find the relationship between study time and marks using the concept of Linear
Regression. Also predict marks for a student if he/she studied for 790 minutes.
Area (ft2) Rent (inr)
360 520
1070 1600
630 1000
890 850
940 1350
500 490
Study Time (min.) Marks obtained
350 520
1070 1600
630 1000
890 850
940 1350
500 490
34. SUMMATIVE ASSESSMENT
34
8. Use the method of Least Square using Regression to predict the final exam grade of a
student who received 86 on mid term exam.
x (midterm) y (final exam)
65 175
67 133
71 185
71 163
66 126
75 198
67 153
70 163
71 159
69 151
9. Create a relationship model for given data to find
relationship between height and weight of students. .
Height (inches) Weight (pounds)
72 200
68 165
69 160
71 163
66 126