The line that maximizes the margin in the associated SVM classification problem is given by B₀ + B₁x + B₂y = 0, where B₀ = 1, B₁ = -1/3, and B₂ = -1/3.
(a) To draw the three points in the Cartesian plane, we plot them according to their respective coordinates:
Point X₁: (2, 1)
Point X₂: (5, 1)
Point X₃: (4, 5)
Now, we label the points as follows:
- X₁ (2, 1) with label y₁ = 1
- X₂ (5, 1) with label y₂ = 1
- X₃ (4, 5) with label y₃ = -1
The graph will show these three points on the plane, with different labels assigned to each point.
Intuitively, the line that maximizes the margin in the associated support vector machine (SVM) classification problem is the line that separates the two classes (y = 1 and y = -1) with the largest possible gap or margin between them. This line should aim to be equidistant from the closest points of each class, maximizing the separation between the classes.
(b) To prove that the guess in part (a) is the unique solution of the optimization problem:
min ||B||² subject to yᵢ(xᵢB + B₀) ≥ 1
We can use the Karush-Kuhn-Tucker (KKT) conditions to derive the solution. The KKT conditions for SVM can be stated as follows:
1. yᵢ(xᵢB + B₀) - 1 ≥ 0 (for all i, the inequality constraint)
2. αᵢ ≥ 0 (non-negativity constraint)
3. αᵢ[yᵢ(xᵢB + B₀) - 1] = 0 (complementary slackness condition)
4. Σ αᵢyᵢ = 0 (sum of αᵢyᵢ equals zero)
Now let's solve the optimization problem for the given dataset and prove that the guess from part (a) is the unique solution.
We have the following points and labels:
X₁: (2, 1), y₁ = 1
X₂: (5, 1), y₂ = 1
X₃: (4, 5), y₃ = -1
Assume the solution for B and B₀ as (B₁, B₂) and B₀.
For point X₁:
y₁(x₁B + B₀) = 1[(B₁ * 2 + B₂ * 1) + B₀] = B₁ * 2 + B₂ + B₀ ≥ 1
This implies: B₁ * 2 + B₂ + B₀ - 1 ≥ 0
For point X₂:
y₂(x₂B + B₀) = 1[(B₁ * 5 + B₂ * 1) + B₀] = B₁ * 5 + B₂ + B₀ ≥ 1
This implies: B₁ * 5 + B₂ + B₀ - 1 ≥ 0
For point X₃:
y₃(x₃B + B₀) = -1[(B₁ * 4 + B₂ * 5) + B₀] = -B₁ * 4 - B₂ * 5 - B₀ ≥ 1
This implies: -B₁ * 4 - B₂ * 5 - B₀ - 1 ≥ 0
Now we can write the Lagrangian function for this optimization problem:
L(B, B₀, α) = (1/2) ||B||² - Σ αᵢ[yᵢ(xᵢB + B₀) - 1]
Using the KKT conditions, we have:
∂L/∂B₁ = B₁ - Σ αᵢyᵢxᵢ₁ = 0
∂L/∂B₂ = B₂ - Σ αᵢyᵢxᵢ₂ = 0
∂L/∂B₀ = -Σ αᵢyᵢ = 0
Substituting the values of xᵢ and yᵢ for each point, we have:
B₁ - α₁ - α₂ = 0
B₂ - α₁ - α₂ = 0
-α₁ + α₂ = 0
Simplifying these equations, we get:
B₁ = α₁ + α₂
B₂ = α₁ + α₂
α₁ = α₂
This implies B₁ = B₂, which means the decision boundary is perpendicular to the vector (1, 1).
Substituting B₁ = B₂ and α₁ = α₂ into the equation ∂L/∂B₀ = -Σ αᵢyᵢ = 0, we get:
-α₁ + α₂ = 0
α₁ = α₂
So, we have α₁ = α₂, which implies that the guess in part (a) is the unique solution for the given optimization problem.
Therefore, the line B₀ + B₁x₁ + B₂x₂ = 0 that maximizes the margin in the associated SVM classification problem is the line perpendicular to the vector (1, 1), passing through the mid-point of the closest points between the two classes.
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xpress the function as the sum of a power series by first using partial fractions. f(x) = 7 x2 − 3x − 10 f(x) = [infinity] n = 0 find the interval of convergence. (enter your answer using interval notation.)
The radius of convergence is 5/7. The interval of convergence is (-5/7, 5/7) in interval notation.
To express the function f(x) = 7x² - 3x - 10 as the sum of a power series, we can start by factoring the quadratic term in the numerator:
f(x) = (7x² - 3x - 10)
The quadratic expression can be factored as follows:
f(x) = (7x + 5)(x - 2)
Now we can write the function f(x) as a sum of partial fractions:
f(x) = A/(7x + 5) + B/(x - 2)
To find the values of A and B, we can multiply both sides of the equation by the denominators and equate the coefficients of corresponding powers of x:
(7x + 5)(x - 2) = A(x - 2) + B(7x + 5)
Expanding both sides of the equation:
7x² - 14x + 5x - 10 = Ax - 2A + 7Bx + 5B
Grouping the terms with the same power of x:
(7x² + (5 - 14)x - 10) = (A + 7B)x + (-2A + 5B)
Equating the coefficients of corresponding powers of x:
7x² + (5 - 14)x - 10 = (A + 7B)x + (-2A + 5B)
Comparing the coefficients:
7 = A + 7B
5 - 14 = -2A + 5B
-10 = -2A
From the first equation, we can solve for A:
A = 7 - 7B
Substituting this value of A into the second equation:
-10 = -2(7 - 7B)
Simplifying:
-10 = -14 + 14B
14B = -10 + 14
B = 4/14
B = 2/7
Now we have the values of A and B:
A = 7 - 7B = 7 - 7(2/7) = 7 - 2 = 5
Therefore, the function f(x) can be expressed as:
f(x) = 5/(7x + 5) + 2/(x - 2)
Now, to find the interval of convergence for the power series representation of f(x), we need to determine the radius of convergence. The power series representation will converge within the interval (-r, r), where r is the radius of convergence.
In this case, since we have a rational function, the interval of convergence will be determined by the denominator with the smallest radius of convergence.
The denominators in the partial fractions are (7x + 5) and (x - 2). The radius of convergence for a power series centered at a point c is the distance from c to the nearest singularity.
For (7x + 5), the singularity occurs when 7x + 5 = 0, which gives x = -5/7.
For (x - 2), the singularity occurs when x - 2 = 0, which gives x = 2.
The distance from the center (c = 0) to the nearest singularity is the minimum of the absolute values of the two singularities: min(|-5/7|, |2|) = 5/7.
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What is the mean:4,3,5,4,8,0 IXL
Which value of x makes the equation -2(1-4x)=3x+8 true?
Answer:
x = 2
Step-by-step explanation:
distribute -2 first
-2 + 8x = 3x + 8
-2 + 5x = 8
5x = 10
x = 2
HELPPP ME PLSSS AND NO BOTS BC I WILL REPORT AND I BARELY HAVE POINT SO PLS HELP ME
Answer: 52 1/2 inches. Draw a vertical line up from the 20 week marker and where that line intersects the slanted red line and. Then take a straight edge and draw a line parallel to the "week" line on the bottom of the graph from the intersection to intersect the height line. Where the second line crosses the height line that number is the height of the plant at 20 weeks.
Andre has a summer job selling magazine subscriptions he earned $25 per week plus $3 for every subscription he sells.Andre hopes to make at least enough money to buy a new pair of soccer cleats
Answer:
Step-by-step explanation:
That Would be 43/3
Simplity (4.5)(5)(-2)
O45
045
0-45
-45
A rod of length 30.0 cm has linear density (mass per length) given by
λ=50.0+20.0 x
where x is the distance from one end, measured in meters, and λ is in grams/ meter.
What is the mass of the rod?
The mass of the rod is 15.9 grams.
To find the mass of the rod, we need to integrate the linear density function over the entire length of the rod. The linear density function is given by λ = 50.0 + 20.0x, where x is the distance from one end measured in meters.
The mass of an infinitesimally small element of length dx is given by dm = λ*dx. Substituting the linear density function, we have dm = (50.0 + 20.0x)*dx.
Integrating both sides from x = 0 to x = 0.3 meters (corresponding to the length of the rod), we get:
∫dm = ∫(50.0 + 20.0x)dx
m = ∫(50.0 + 20.0x)dx
m = [50.0x + 10.0x^2] evaluated from x = 0 to x = 0.3
m = 50.00.3 + 10.0(0.3)^2
m = 15.0 + 0.9
m = 15.9 grams.
Therefore, the mass of the rod is 15.9 grams.
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can anyone help me on this question but explain? What’s the difference between gross and net income
Answer & Explanation:
Gross income is the total pay you earn before any reduction and adjustments (taxes, mortgage, etc.).
Net income is the amount of pay you get after taxes and other reductions are taken out of your gross income; it's the money that you "actually get to take home in your pocket with" on the payday.
b+5b+6b 6a-4a 10a+3a+2a 9p+20p+8p 12x -10y 2+5x+3+5x Упрости выражения,где это возможно. ПОМОГИТЕ ПРОШУ!!!!!!!
Answer:
i dont undersant what u r trying to say
Step-by-step explanation:
i thnk this is harsdath
Ms. Thompson went to buy socks for her Jordans. She brought $20 to the store. Each pair of scold costs $2.50. How many pairs of socks can she buy with $20?
answer of this question:
8
Step-by-step explanation:
$20÷$2.50
=8
:)
Help me really please asp
Answer:
y1=4
x2=7
y2=-3
if a club has 20 members and 4 officers how many choices are there for a secretary
Answer:
20 x 4 = 60
Step-by-step explanation:
because it's says how many
Answer:
18 maybe
Step-by-step explanation:
it's either 20,19,or 18
but the secretary is the third to be elected so the answer is 18 maybe and if that doesn't work try one of the others trust me it's one of them.
hope this helps
if f(5) = 11, f ′ is continuous, and 6 5 f ′(x) dx = 19, what is the value of f(6)?
If f(5) = 11, f ′ is cοntinuοus, and 6 5 f ′(x) dx = 19, then f(6) = 30.
What is Fundamental Theοrem οf Calculus?Tο find the value οf f(6), we can use the Fundamental Theοrem οf Calculus. Accοrding tο the theοrem, if f(x) is cοntinuοus οn an interval [a, b] and F(x) is an antiderivative οf f(x) οn that interval, then the definite integral οf f(x) frοm a tο b is equal tο F(b) - F(a).
Given that f'(x) is cοntinuοus, we can apply the theοrem tο the integral:
∫₅₆ f'(x) dx = f(6) - f(5)
We are given that ∫₅₆ f'(x) dx = 19, and f(5) = 11. Plugging in these values, we have:
19 = f(6) - 11
Tο sοlve fοr f(6), we add 11 tο bοth sides:
f(6) = 19 + 11
f(6) = 30
Therefοre, the value οf f(6) is 30.
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Using hypothesis testing, determine whether the sample mean is not equal to the block population's mean (R+) with a confidence level of 99%.
Hypothesis testing is a statistical method used to determine if a hypothesis regarding a population parameter is correct or not.
It is a decision-making process that aids in making decisions about population parameters when only a sample statistic is available. It has the following steps: State the null and alternative hypotheses. Choose the significance level. Determine the critical value or p-value. Calculate the test statistic. Make a decision and state the conclusion. The formula for the test statistic is given, where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. The null and alternative hypotheses for this problem are:H0: μ = R+ (the sample mean is equal to the block population's mean)Ha: μ ≠ R+ (the sample mean is not equal to the block population's mean)We will use a two-tailed test since we are testing whether the sample mean is not equal to the block population's mean.
The significance level is given as 99%. This means that α = 1 - 0.99 = 0.01.The critical value for a two-tailed test with α = 0.01 and degrees of freedom (df) = n - 1 is obtained from a t-distribution table. Since the sample size is not provided, we cannot determine the critical value. The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. The p-value for a two-tailed test is given by:
P-value = P(|t| > |t*|)where t* is the test statistic and |t| is the absolute value of the test statistic. Since we do not have the sample size or the test statistic, we cannot calculate the p-value. Therefore, we cannot make a decision and state a conclusion about the hypothesis test.
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16 is what percent of 25
Answer:
4
Step-by-step explanation:
Answer:
64 %
Step-by-step explanation:
( 16 / 25 ) x 100
= ( 16 x 100 ) / 25
= 16 x 4
= 64 %
simplify -2+3(1-4)-2
Answer:
-13
Step-by-step explanation:
−2+3(1−4)−2
=−2+(3)(−3)−2
=−2+−9−2
=−11−2
=−13
hope it helped
please mark me as brainliest.
200% of what number is 350?
Answer:
The answer is 175
What is the simple interest earned on an $4,350 investment for 5 years at a rate of 2%?
Hey!
A = $4,785.00
I = A - P = $435.00
Equation:
A = P(1 + rt)
Calculation:
First, converting R percent to r a decimal
r = R/100 = 2%/100 = 0.02 per year.
Solving our equation:
A = 4350(1 + (0.02 × 5)) = 4785
A = $4,785.00
The total amount accrued, principal plus interest, from simple interest on a principal of $4,350.00 at a rate of 2% per year for 5 years is $4,785.00.
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Hope you have a great day and this helped in some way <33
The length of a rectangle is 3ft less than double the width, and the area of the rectangle is 14ft^2 find the dimensions of rectangle.
Answer: [tex]4\times 3.5\ ft^2[/tex]
Step-by-step explanation:
Given
area of the rectangle is [tex]A=14\ ft^2[/tex]
Suppose width is given by w
So, length is [tex]l=2w-3[/tex]
The area of a rectangle is [tex]A=l\times w[/tex]
putting values
[tex]\Rightarrow A=(2w-3)w\\\Rightarrow 14=2w^2-3w\\\Rightarrow 2w^2-3w-14=0\\\Rightarrow 2w^2-7w+4w-14=0\\\Rightarrow w(2w-7)+2(2w-7)=0\\\Rightarrow (2w-7)(w+2)=0\\\\\Rightarrow w=\dfrac{7}{2}\ ft[/tex]
length is [tex]l=4\ ft[/tex]
guys i need help ASAP!!!
Answer:
Q7 0.2
Q8 0.17
Step-by-step explanation:
Probability is the ratio of the number of possible outcome to the number of total outcome. The probability that an event will happen when added to the probability that it will not happen is 1
Given that the probability that the bulb will grow is 0.8, the probability it would not grow
= 1 - 0.8
= 0.2
A match may be won lost or drawn
Given that the probability of a win is 0.28 and that of a draw is 0.55, the probability of a loss
= 1 - (0.28 + 0.55)
= 0.17
What is the mode of the set of numbers?
2, 2, 3, 4, 4, 5, 7, 7, 7, 10, 16, 17
Answer:
It is 7
Step-by-step explanation:
Determine whether the given differential equation is exact. If it is exact, solve it. (If it is not exact, enter NOT.) dy/ dx = 9xe^x – y + 6x2
The given differential equation is not exact. To determine if a differential equation is exact, we need to check if its partial derivatives satisfy a specific condition.
In this case, the given equation is dy/dx = 9xe^x – y + 6x^2. Let's find the partial derivatives of the expression on the right-hand side with respect to y and x.∂/∂y (9xe^x – y + 6x^2) = -1∂/∂x (9xe^x – y + 6x^2) = 9e^x + 12x
The condition for exactness is that these partial derivatives are equal: ∂/∂y (∂/∂x) = ∂/∂x (∂/∂y). However, in this case, we have -1 ≠ 9e^x + 12x, which means the equation is not exact.As the equation is not exact, we cannot directly solve it by finding a potential function. Other methods, such as using integrating factors or applying specific techniques for solving non-exact equations, may be required.
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i light is faster then anything how did dark get there first
hello there enjoy 30 points on da house
Answer:
this just confused me so much
Step-by-step explanation:
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4
Select the correct answer.
Solve the following equation for x.
x2 - 9x+ 18 = 0
O A.
X = -3; x = 6
ОВ.
X= 3; x = 6
OC.
x= -3; x = -6
OD.
x= 3; x = -6
(行
Reset
Next
Answer:
B
Step-by-step explanation:
Given
x² - 9x + 18 = 0 ← in standard form
(x - 3)(x - 6) = 0 ← in factored form
Equate each factor to zero and solve for x
x - 3 = 0 ⇒ x = 3
x - 6 = 0 ⇒ x = 6
solution is x = 3, x = 6 → B
A 13 foot ladder is leaning against the wall of a house. The ladder is 5 feet away from th
base of the house. How high up the house does the ladder reach?
Answer:
12 feet
Step-by-step explanation:
a car can travel 174 miles in 4 hours how many miles can the car travel per hour
Answer:
43.5
divide 174 ÷ 4 and you get 43.5
Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. h(x)=e−3x;h(13),h(1.5),h(−1),h(−π)h(x)=e−3x;h(31),h(1.5),h(−1),h(−π)
Given the function h(x) = e-3x and we have to evaluate the function at the indicated values. So, we need to substitute the values in the given function and find the value of the function at the given points as shown below: h(13) = e-3(13)h(13) = e-39h(13) = 0.000027323h(1.5) = e-3(1.5)h(1.5) = e-4.5h(1.5) = 0.01111h(-1) = e-3(-1)h(-1) = e3h(-1) = 20.08554h(-π) = e-3(-π)h(-π) = e3πh(-π) = 23.14069.
(a) For, h(-1):
Plug x = -1 into the function:
h(-1) = e^(-3*-1)
Using a calculator, we find that h(-1) ≈ 20.086.
Similarly,
(b) For, h(-π):
Plug x = -π into the function:
h(-π) = e^(-3*-π)
Using a calculator, we find that h(-π) ≈ 23.141.
So, the value of h(x) at the given values are : h(13) = 0.000027323, h(1.5) = 0.01111, h(-1) = 20.08554 and h(-π) = 23.14069.
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An economist conducted a study of the possible association between weekly income and weekly grocery expenditures. The particular interest was whether higher income would cause shoppers to spend more en groceries. A random sample of shoppers at a local supermarket was obtained. A questionnaire was administered asking about the weekly income of each shopper's family and their grocery bill for that werk. The gender of each shopper was also obtained. The data below are expenditures and income for 10 selected survey participants. Income Grocery 98 52 201 78 298 108 398 95 481 198 600 99 738 162 805 187 890 105 1023 173 The correlation for these data is given by 0.794 Ob-0.619. 0.649 4.0.735.
The correlation coefficient for the data is 0.794.
The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient of 0.794 indicates a strong positive correlation between weekly income and weekly grocery expenditures.
A correlation coefficient value of 0.794 suggests that as the weekly income increases, the weekly grocery expenditures tend to increase as well. The positive correlation implies that shoppers with higher incomes tend to spend more on groceries.
It is important to note that correlation does not imply causation. The study observed a correlation between income and grocery expenditures, but it does not necessarily mean that higher income directly causes shoppers to spend more on groceries. Other factors and variables may also influence grocery spending habits.
In summary, based on the given data, there is a strong positive correlation (0.794) between weekly income and weekly grocery expenditures for the surveyed shoppers.
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A spherical globe has a diameter of 10 inches. What is the approximate volume of the globe?
Answer:
523.33 in^3
Step-by-step explanation:
A sphere is a 3-dimensional version of a circle. An example of a sphere is a ball
Volume of a sphere = [tex]\frac{4}{3}[/tex]π[tex]r^{3}[/tex]
Where
π = 3.14
r = radius
the diameter is the straight line that passes through the centre of a circle and touches the two edges of the circle.
A radius is half of the diameter
diameter = 2r
Radius = 10 in / 2 = 5 inches
[tex]\frac{4}{3}[/tex] × 3.14 ×[tex]5^{3}[/tex] = 523.33 [tex]in^{3}[/tex]
Find the Perimeter of the figure below, composed of a rectangle and two
semicircles. Round to the nearest tenths place.
10
HELP MEHHH....pls
Answer:
Step-by-step explanation: