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CS 165 > Optimisation and Regression > Flashcards

Optimisation and Regression Flashcards

1
Q

What are independent variables

A

Independent variables are input variables that can be freely changed or manipulated, e.g. age of a person

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2
Q

What are dependent variables

A

Dependent variables are output that cannot be changed freely without altering the inputs, e.g. temperature of a room after adjusting the thermostat

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3
Q

What are categorical variables

A

A categorical variable is a characteristic that in not quantifiable, e.g. type of fruit

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4
Q

What are nominal variables

A

A nominal variable is a categorical variable where there is no natural order or hierarchy, e.g. gender of person

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5
Q

What are ordinal variables

A

An ordinal variable is a categorical variable where there is a natural order or hierarchy, e.g. Education level

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6
Q

What are numeric variables

A

A numeric variable is a characteristic that is quantifiable, e.g. Number of items sold

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7
Q

What are continuous variables

A

A continuous variable is a numeric variable that takes real values of infinite precision, e.g. Time taken to complete a task (in seconds)

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8
Q

What are discrete variables

A

A discrete variable is a numeric variable that takes finite options, e.g. number of cars in a car park

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9
Q

What is a function

A

A function defines the relationship between inputs and output(s)

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10
Q

How is a function like a mathematical model

A

A function is like a mathematical model because it tells how the output(s) would vary given a change in inputs

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11
Q

What is the definition of optimization

A

Optimization is the process of finding the best option or solution form a set of alternatives

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12
Q

What is the goal of optimization

A

The goal of optimization is to find a specific vector of input or independent variables that produce a desired output or dependent variable value

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13
Q

What are the input variables in optimization

A

The input variables in optimization are the independent variables, which can be free changed or manipulated

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14
Q

What is the dependent variable in optimization

A

The dependent variable in optimization is the output, which cannot be changed freely without altering the inputs

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15
Q

What is the objective of optimization

A

The objective of optimization is to maximize or minimize a particular objective function, which is a mathematical expression that represents the relationship between the input and output variables

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16
Q

What are some examples of optimization problems

A

Finding the most profitable investment portfolio given a set of assets and market conditions
Designing an aeroplane wing that minimizes drag and maximise lift
Determining the optimal production schedule for a manufacturing plant to minimize costs and maximise profits

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17
Q

What are some common techniques used in optimization

A

Trial and Error
Geometric
Metaheuristics
Data-driven

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18
Q

What is the trial and error approach

A

The trial and error approach involves trying different solutions and observing the outcome to find the best option

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19
Q

What is the geometric approach

A

The geometric approach involves generating new points using some form of geometric knowledge, such as rotating or reflecting a point

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20
Q

What is the calculus approach

A

The calculus approach involves evaluating the function and its derivatives to direct the search in the direction that minimises or maximises the function

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21
Q

What are metaheuristics

A

Metaheuristics are problem-solving techniques that draw inspiration from natural processes to find solutions. Examples include simulated annealing, genetic algorithms, and particle swarm optimisation

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22
Q

What is the data-driven approach

A

The data-driven approach involves using information gathered from previous solutions to improve search for the best option. This can include machine learning or statistical modelling to make predictions based on past data

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23
Q

When should you use trial and error

A

When the problem is simple, and the number of possible solutions is limited. It is also appropriate when the cost of failure is low and the outcome of each trial can be easily observed

24
Q

When should you use geometric

A

When the problem involves finding an optimal geometric arrangement or position. It is also appropriate when the optimization problem can be formulated as a geometric optimization problem, such as finding the shortest distance between two points

25
Q

When should you use calculus

A

When the optimization problem can be formulated as a mathematical function and the function and its derivatives can be easily evaluated. It is also appropriate when the optimization problem involves finding the maximum or minimum value of a continuous function

26
Q

When should you use metaheuristics

A

When the optimization problem is complex and the search space is large and/or multi-dimensional. Metaheuristics can be used when the optimization problem is non-linear or when the function to be optimized is not known

27
Q

When should you use data-driven

A

When there is a large amount of data available and patterns can be extracted from the data. It is also appropriate when the optimization problem is complex and the relationship between the input and output variables is not well-understood

28
Q

Is finding the global optimum always feasible in optimization

A

No, finding the global optimum can be difficult or even impossible, especially when dealing with complex problems with a large number of variables and constraints

29
Q

What is a local optimum

A

A local optimum is the best solution within a certain region of the search space

30
Q

Is a local optimum always the best

A

No, a local optimum may not always be the best solution for the overall problem

31
Q

How can local optima be found

A

Local optima can be found using various optimization algorithms, such as geometrics or metaheurestics

32
Q

Why are we usually happy with a solution that is “good enough”

A

Finding the global is often too difficult, so we settle for a solution that is “good enough” to meet requirements of the problem

33
Q

What is regression

A

Regression is the process of predicting a premise value for the output (or dependent) variable based on a value of the input (or independent) variable

34
Q

What is linear regression

A

Linear regression is a type of regression where the relationship between the input variable(s) and the output variable is modelled as a straight line

35
Q

What are the characteristics of a line in linear regression

A

A line in linear regression is straight with no bends, has no thickness, and extends to positive or negative infinity

36
Q

What is the mathematical function for a line in linear regression

A

y = mx + c, where y is the output variable, x is the input variable, m is the slope of the line, and c is the y-intercept of the line

37
Q

How is linear regression used for prediction

A

In linear regression, the input variable(s) are used to predict the output variable by finding the best-fitting line that minimizes the distance between predicted values and the actual values. Once the line is established, the value of the output variable can be predicted for any given value of the input variable

38
Q

How do you generalise a linear equation for 1-D input space

A

y = f(x,m,c) = mx + c

39
Q

How do you generalise a linear equation for 2-D input space

A

y = f(x0, x1, m0, m1, c) = m0x0 + m1x1 + c

40
Q

How do you generalise a linear equation for N-D input space (Check notes)

A

y = f(x0, x1, …, xn-1, m0, m1, …, mn-1, c) = m0x0 + m1x1 + … + mn-1xn-1 + c

41
Q

What does the error or residual represent in a linear regression context

A

The error on residual represents the difference between the predicted value and the actual value of the dependent variable for a particular data point in linear regression

42
Q

How do we calculate the error or residual for a single data point

A

To calculate the error or residual for a single data point, we subtract the predicted value from the actual value of the dependent value

43
Q

Why do we square the error or residual in linear regression

A

We square the error or residual in linear regression to avoid cancellation of positive and negative errors and to emphasize larger errors, which helps in identifying and reducing them

44
Q

What does a negative error or residual value indicate in linear regression

A

A negative error or residual value indicates that the predicted value is higher than the actual value of the dependent variable for a particular data point in linear regression

45
Q

What is the error or residual value in the given example (check notes), where the estimated value is y = 1000 and the measured value is y = 1025

A

The error or residual value is e = 1025 - 1000 = 25, and the squared error or residual is e^2 = 625

46
Q

What is the model estimate

A

The model estimate is the predicted value of the output variable based on the input variable(s)

47
Q

What is the true measurement

A

The true measurement is the actual value of the output variable

48
Q

General Error Function (for one point)

A

Check notes

49
Q

General Error Function (for k points)

A

Check notes

50
Q

General Error Function (for k points)

A

Check notes

51
Q

What is a neuron

A

A neuron is the basic unit of a brain that works as an information messenger

52
Q

How does a neuron work

A

A neuron receives electrical impulses and chemical signals through its axon and forwards them - if it is excited enough - to other neurons through its dendrites

53
Q

What happens when neurons pass messages to each other

A

The simple message passing between neurons can lead to complex behaviours and functions

54
Q

What is the role of axons and dendrites in neurons

A

Axons are responsible for transmitting electrical impulses and chemical signals away from the neuron’s cell body, while dendrites receive impulses from other neurons and transmit them towards the neuron’s cell body

55
Q

How are neurons related to artificial neural networks

A

Artificial neural networks are inspired by the structure and function of biological neurons, and use mathematical models of neurons to perform tasks such as classification, prediction, and control

56
Q

READ YOUR NOTES FOR IMAGES

A

NOW

CS 165 (8 decks)

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