text:
## Exercises - Descriptive Statistics

1. Given the following data: 100, 95, 95, 90, 85, 75, 65, 60, 55. Find the median, mean, and mode. Is there a most appropriate measure?

median = 85, mean = 80, mode = 95; The mode is certainly inappropriate. Beyond that, the data set is too small to know whether the mean or median is more appropriate.

2. Make a sketch of the following, indicating the approximate locations for the mean, median and mode:

1. a normal distribution
2. a skewed distribution
3. a rectangular distribution

See notes.

3. Given the data set: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ''x''. Find the smallest positive integer value for ''x'' such that ''x'' is an outlier. Find the value for ''both'' definitions.

Experiment to find these values.

4. Given the following set of golf scores: 67, 70, 72, 74, 76, 76, 78, 80, 82, 85. Find the median, mean, mode, and standard deviation. What percentage of scores are in the interval of one standard deviation from the mean?

median = 76, mean = 76, mode = 76, standard deviation = 5.52; 6 out of 10 scores are between 70.48 (76 - 5.52) and 81.52 (76 + 5.52), giving 60% of scores for this data set within one standard deviation of the mean

5. Give at least five uses and at least five misuses of statistics.

6. Give the four levels or categories of data and give an example of each.

7. Given the following grades on a test: 86, 92, 100, 93, 89, 95, 79, 98, 68, 62, 71, 75, 88, 86, 93, 81, 100, 86, 96, 52

1. A stem-and-leaf plot is a quick way to construct a histogram by hand, whereby one lumps together all values that have common leading digits up to some level of precision (i.e., the "stem") using the remaining digit(s) of each data value to form the "bars" (i.e., the "leaves"), as shown below for the data set $13,15,17,17,19,24,25,25,25,26,29,30,31,33,46,62$:

6 | 2
5 |
4 | 6
3 | 0 1 3
2 | 4 5 5 5 6 9
1 | 3 5 7 7 9

Make a stem-and-leaf plot that represents the test grade data.
10 | 0 0
9 | 2 3 3 5 6 8
8 | 1 6 6 6 8 9
7 | 1 5 9
6 | 8
5 | 2

2. Find the mode, median, mean, range, standard deviation, and interquartile range

mode = 86, median = 87, mean = 84.5, range = 52 to 100, standard deviation = 13.17, interquartile range (IQR) = 77 to 94
In R:

grades = c(86, 92, 100, 93, 89, 95, 79, 98, 68, 62, 71, 75, 88,
86, 93, 81, 100, 86, 96, 52)

median(data)
mean(data)
range(data)
sd(data)
IQR(data)

# Sadly, the mode() function in R does something else.
# So you can write your own function, or if the data
# set is small enough (as it is here), you can use
# sort(data) to make it easier to count duplicate entries

TI-83:

Enter data in L1 with [STAT] : EDIT : Edit...
Then [STAT] : CALC : 1-Var Stats

This returns a scrollable list with:
* median given by "Med"
* mean given by x
* range given by minX and maxX (technically, their difference)
* standard deviation given by Sx (presuming the data is a sample)
* interquartile range given by Q3 - Q1

3. Are there any outliers? Explain clearly.

For a score to be an outlier, the score must be outside the interval of three standard deviations from the mean (44.99 to 124.01). There is not an outlier.

8. What is an experimental design and why is it important? Describe a completely randomized experimental design and a rigorously controlled design.

See the notes.

9. Given this sample of freshman GPA scores:

2.2, 2.9, 3.5, 4.0, 3.9, 3.5, 2.9, 2.8, 3.1, 3.5, 3.8, 4.0,
3.8, 2.4, 3.9, 3.4, 2.8, 2.4, 1.8, 3.6, 3.1, 2.9, 3.8, 4.0

1. Is there an outlier? (Check both tests and explain)

1.8 is NOT an outlier, as 1) it is not below 1.35 (i.e., the mean minus three standard deviations); and 2) It is not below 1.425 (Q1 minus 1.5 times 0.95).

2. Draw a frequency histogram using 5 to 6 categories. Be consistent with the rules for making histograms.

One set of boundaries includes: 1.65 to 2.05; 2.05 to 2.45; 2.45 to 2.85; 2.85 to 3.25; 3.25 to 3.65; 3.65 to 4.05, as shown below -- although a variety of answers will work.

In R:
data = c(2.2, 2.9, 3.5, 4.0, 3.9, 3.5, 2.9, 2.8, 3.1, 3.5, 3.8, 4.0,
3.8, 2.4, 3.9, 3.4, 2.8, 2.4, 1.8, 3.6, 3.1, 2.9, 3.8, 4.0)

hist(data,
breaks=seq(from=1.65,to=4.05,by=0.40),
col="gray",
xlab="GPA",
main="GPA Scores")

3. Is the distribution significantly skewed?

Pearson's Index gives a value of $-0.95$, so it is not significantly skewed (although the distribution looks skewed from the histogram)

format: plain


text:
MATLAB function printing extra matrix?

• MATLAB

Main Question or Discussion Point

I'm a newbie to all things programming, and have to write a program implementing the Euler method to approximate an ODE.

Anyways, the program works but when I use the function I wrote I get an extra matrix of zeroes above the desired output, which I don't understand as I have suppressed the output of the zero matrix with a semi-colon - the matrix is Eulerpoints. Here's the source code:

function [Eulerpoints] = Eulermethod(x0,y0,xn,steps)
%EULERMETHOD - this function will approximate y by the Euler method.
% EULERMETHOD(x0,y0,xn,steps) will return a list of points that
% approximate y in the interval [x0,xn] given our initial co-ordinate
% (x0,y0) and the number of steps.

Eulerpoints=zeros(steps+1,2);
start = 2;
finish = steps+1;
h=(xn-x0)/steps;
Eulerpoints(1,1) = x0, Eulerpoints(1,2) = y0;
fprintf(' n xn Yn\n 0 %2g %2g\n',x0,y0)

for I= start:finish
Eulerpoints(I,1) = x0 +h*(I-1);
Eulerpoints(I,2) = Eulerpoints(I-1,2) + h*(-2*Eulerpoints(I-1,2) - 0.25*Eulerpoints(I-1,1)*Eulerpoints(I-1,1) + 0.125); %% This line contains the ODE, which can be adjusted for a different equation.
fprintf('%3g %4g %6g\n',I-1,Eulerpoints(I,1),Eulerpoints(I,2))
end
%
end

Related MATLAB, Maple, Mathematica, LaTeX News on Phys.org
The x0 in Line 11 needs to have a semi-colon after it. This will then return the Eulerpoints matrix as a variable and the [n xn yn] print out that you call for, but you will no longer have the output you were concerned with.

Here is the fixed code (using the CODE wrap that we have here on PF):

Code:
function [Eulerpoints] = Eulermethod(x0,y0,xn,steps)
%EULERMETHOD - this function will approximate y by the Euler method.
% EULERMETHOD(x0,y0,xn,steps) will return a list of points that
% approximate y in the interval [x0,xn] given our initial co-ordinate
% (x0,y0) and the number of steps.

Eulerpoints=zeros(steps+1,2);
start = 2;
finish = steps+1;
h=(xn-x0)/steps;
Eulerpoints(1,1) = x0; Eulerpoints(1,2) = y0;
fprintf(' n xn Yn\n 0 %2g %2g\n',x0,y0)

for I= start:finish
Eulerpoints(I,1) = x0 +h*(I-1);
Eulerpoints(I,2) = Eulerpoints(I-1,2) + h*(-2*Eulerpoints(I-1,2) - 0.25*Eulerpoints(I-1,1)*Eulerpoints(I-1,1) + 0.125); %% This line contains the ODE, which can be adjusted for a different equation.
fprintf('%3g %4g %6g\n',I-1,Eulerpoints(I,1),Eulerpoints(I,2))
end
%
end
Hope this helps.

Ben

Ah thanks, yeh that's been bugging me (geddit?) for ages.
Cheers


text:
# P2255 [USACO14JAN]记录奥林比克Recording the M…

• 495通过
• 1K提交
• 题目提供者 FarmerJohn2
• 评测方式 云端评测
• 标签 动态规划,动规,dp 排序 贪心 USACO 2014
• 难度 普及+/提高
• 时空限制 1000ms / 128MB
• 提示：收藏到任务计划后，可在首页查看。

## 题目描述

Being a fan of all cold-weather sports (especially those involving cows),Farmer John wants to record as much of the upcoming winter Moolympics as possible.

The television schedule for the Moolympics consists of N different programs(1 <= N <= 150), each with a designated starting time and ending time. FJ has a dual-tuner recorder that can record two programs simultaneously.

Please help him determine the maximum number of programs he can record in total.

农民约翰热衷于所有寒冷天气的运动（尤其是涉及到牛的运动）， 农民约翰想录下尽可能多的电视节目。 为moolympics电视时间表由N个不同的节目 （1 < = n＝150），每个具有指定的开始时间和结束时间。FJ 有一个双调谐器录音机，可以同时录制两个节目。 请帮助他确定他能录制的节目的最大数量。

## 输入输出格式

输入格式：

* Line 1: The integer N.

* Lines 2..1+N: Each line contains the start and end time of a single program (integers in the range 0..1,000,000,000).

第1行：整数n。 第2行.. 1 + n：每行包含单个的开始和结束时间 程序（范围为0…1000000000的整数）。

输出格式：

* Line 1: The maximum number of programs FJ can record.

仅一行，节目FJ可以记录的最大数量。

## 输入输出样例

输入样例#1： 复制
6
0 3
6 7
3 10
1 5
2 8
1 9
输出样例#1： 复制
4

## 说明

INPUT DETAILS:

The Moolympics broadcast consists of 6 programs. The first runs from time 0 to time 3, and so on.

OUTPUT DETAILS:

FJ can record at most 4 programs. For example, he can record programs 1 and 3 back-to-back on the first tuner, and programs 2 and 4 on the second tuner.

Source: USACO 2014 January Contest, Silver

提示
标程仅供做题后或实在无思路时参考。
请自觉、自律地使用该功能并请对自己的学习负责。
如果发现恶意抄袭标程，将按照I类违反进行处理。


text:
# GIT quotients for linear representations of $SL(2,\mathbb C)$

Let $V$ be the standard two-dimensional representation of $SL(2,\mathbb C)$ and let ${\rm Sym}^2V$ be its symmetric square. Let $n$ be a positive integer and consider the following two representations 1) $V^{\oplus n}$ and 2) $({\rm Sym}^2V)^{\oplus n}$ of $SL(2,\mathbb C)$.

Question 1) Is there some explicit description of the GIT quotient $V^{\oplus n}// SL(2,\mathbb C)$? In particular, is it true that the ring of invariant polynomials is generated by $\frac{n(n-1)}{2}$ quadratic polynomials $vol(v_i,v_j)$ $i\ne j$, where $v=(v_1,\cdots ,v_n)=v\in V^{\oplus n}$ and $vol$ is a volume preserved by $SL(2,\mathbb C)$?

Question 2). Is there some explicit description of the GIT quotient $({\rm Sym}^2V)^{\oplus n}// SL(2,\mathbb C)$? In particular, is it true that the ring of invariant polynomials is generated by $\frac{n(n+1)}{2}$ quadratic polynomials $(v_i,v_j)$ , where $v=(v_1,\cdots ,v_n)=v\in ({\rm Sym}^2V)^{\oplus n}$ and $(.,.)$ is a symmetric bilinear form on $({\rm Sym}^2V)$ preserved by $SL(2,\mathbb C)$?

• Mumford's description of the geometric quotient of $V^{\oplus n}$ by the standard action of $\text{SL}(V)$ in Chapter 3 of GIT is so very explicit that it does not actually use any of the general theory of "Geometric Invariant Theory"! That is important, because Mumford wanted to construct $\mathcal{M}_g$ as a scheme over $\text{Spec}(\mathbb{Z})$, whereas GIT (at that time) only worked over a fixed field. So Mumford uses his explicit quotient construction of $V^{\oplus n}$ over any base as a step in constructing $\mathcal{M}_g$ as a quasi-projective scheme over $\text{Spec}(\mathbb{Z})$. – Jason Starr Apr 4 '17 at 15:06

Both questions are extensively dealt with in Weyl's book "Classical invariant theory" who investigated the invariant of classical groups on multiple copies of their defining representations. Determining a set of generators is called a "First Fundamental Theorem" (FFT) while the relations are given in a "Second Fundamental Theorem".

Question 1: This is best regarded as the action of $Sp(2n)$ on $V=\mathbb C^{2n}$ for $n=1$. Then, indeed, the ring of invariants is generated by all pairings $f_{ij}:=\omega(v_i,v_j)$. These are neatly organized in a $2n\times 2n$ skew-symmetric matrix. The relations are generated by all "principal" Pfaffian minors of size $(2n+2)\times (2n+2)$. For $n=2$ these are quadratic polynomials in 3 variables called the "Plücker relations". The quotient consists of all skew-symmetric matrices of rank $\le 2$ (which is, of course, the affine cone over a Grassmannian).

Question 2: In this case one is dealing with the group $SO(n)$ acting on $\mathbb C^n$ for $n=3$. Here things are a bit more complicated since $SO(n)$ is not strictly a classical group. The better problem is to look at the group $O(n)$ instead. In this case, the invariants are indeed generated by all pairings $p_{ij}=(v_i,v_j)$ which can be organized into a $n\times n$ symmetric matrix. The relations are generated by all $(n+1)\times(n+1)$-principal minors. In your case, the quotient would be the set of symmetric matrices of rank $\le3$. Since you are dealing with $SO(n)$ instead of $O(n)$ things are more complicated. In this case there are additional generating invariants namely all determinants of the form $\det(v_{i_1},\ldots,v_{i_n})$. For $n=3$ the quotient is the subset of $S^2\mathbb C^n\oplus\wedge^3\mathbb C^n$ given be two sets of relations: the rank conditions and the condition that the square of the determinant can be expressend as a Gram matrix $\det((v_{i_\mu},v_{i_\nu}))$.

• Thank you! I imagine this is contained in your answer, but is it possible to spell out a bit more explicitly what is the GIT quotient a cone over, in the second case? (if it is again a cone over something) – aglearner Apr 4 '17 at 20:20
• It is not a space one encounters in a Linear Algebra course. Varieties like that have been studied. Search for "determinantal varieties". It carries a $GL(n,\mathbb C)$-action which makes it into a spherical variety with probably 3 orbits. It is a singular with rational singularities (in particular normal and Cohen-Macaulay). – Friedrich Knop Apr 5 '17 at 5:41
• Friedrich, many thanks! I have not realized that this is a spherical variety. I am not worried too much that this variety is not from a linear algebra course :). But I am not able to understand the last three lines of your answer. What are "the rank conditions" and the condition that "the square of the determinant can be expressed as a Gram matrix" ? – aglearner Apr 5 '17 at 9:06
• In fact, I understand that there is a morphism from $\rm Sym^2(V)^{\oplus n}//SL(2,\mathbb C)$ to the cone over the Grassmanian $G(3,n)$. For a generic point the fiber is $6$-dimensional. But I don't see how to get an action with one open orbit on $\rm Sym^2(V)^{\oplus n}//SL(2,\mathbb C)$... So I don't understand what is the spherical variety in this case... – aglearner Apr 5 '17 at 11:38

For question 1, the answer is most easily described by thinking about $V^{\oplus n}$ as $Hom(\mathbb C^2, \mathbb C^n)$. From this it follows that the projective GIT quotient $$V^{\oplus n} //_{det} GL_2$$ is isomorphic to the Grassmannian $\mathbb G(2,n)$. From this, it follows that the invariant ring $\mathbb C[V^{\oplus n}]^{SL_2}$ is homogeneous coordinate ring of $\mathbb G(2,n)$. Your quadratic polynomials are the generators of this coordinate ring (and the relations are given by the Plucker relations).

• I suppose the FFT is hidden in the isomorphism with the Grassmannian. – Abdelmalek Abdesselam Apr 4 '17 at 17:58

Question 1: yes.

Question 2: no.

Explicit linear generators follow from the first fundamental theorem (FFT) for $SL_2$. You can see my two answers to this MO question for an explicit proof of the FFT for $SL_k$. It is best to use a graphical language to represent these generators as in my article "On the volume conjecture for classical spin networks". J. Knot Theory Ramifications 21 (2012), no. 3, 1250022. This type of graphical representation is very old as you can see in this MO answer. Then the fun begins, namely trying to find polynomial rather than linear generators. Essentially this results from the Grassmann-Plücker relation. For $SL_2$ and forms of degree 1 or 2 this is easy to do by hand. For quadratics, the GP relation can be used to break cycles (the only thing produced by the FFT). In fact the article by Kempe in the second MO answer I mentioned does exactly that, with explanatory pictures. The first invariant which is not expressible by the ones you gave is for three quadratics corresponding to a 3-cycle containing each one of them. This is also the Wronskian of the three quadratics.

For quadratics an explicit system of generators which basically adds these Wronskians for each triple of forms is given in Section 256 "The quadratic types" in the book by Grace and Young.

The basic identity from that book needed for breaking cycles of length at least four is in classical symbolic notation: $$2(ab)(bc)(cd)(de)=(bc)(cd)(db)(ae)-(cd)^2(ab)(be)-(bc)^2(ad)(de)+(bd)^2(ac)(ae)$$ where $a=(a_1,a_2)$ etc. and $(ab)$ is the determinant of the matrix with first row $a$ and second row $b$ etc.

Using self-duality of $SL_2$ representations, the LHS can be a interpreted as a product of four $2\times 2$ matrices. This is basically the Amitsur-Levitzki Theorem for $2\times 2$ matrices.

I don't know how explicit you want to be but you can look at the article "Defining Relations for the Algebra of Invariants of 2×2 Matrices" by Drensky for more details. He treats the case of generic matrices while you are interested in matrices coming from symmetric bilinear forms by self-dualization. For generic matrices, traces of words of length 1, 2, 3 form a minimal system of $n+\frac{n(n-1)}{2}+\frac{n(n-1)(n-2)}{6}$ algebra generators. Drensky finds the polynomial relations between these generators. Here, with quadratic binary forms, the words of length 1 disappear.

• Thank you! I guess the article of Drensky indeed addresses the second half of the question. I would be happy if there was some "geometric" description of this GIT quotient akin to the cone over Grassmannian $G(2,n)$ appearing in the part 1) of the question. (But it might be that such a description does not exist...) – aglearner Apr 5 '17 at 3:35


text:
# 4.3: Directions and Magnitudes

• Page ID
1857
•

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Consider the $$\textit{Euclidean length}$$ of a vector:

$\|v\| := \sqrt{(v^{1})^{2} + (v^{2})^{2}+\cdots(v^{n})^{2}}\ =\ \sqrt{ \sum_{i=1}^{n} (v^{i})^{2} }\: .$

Using the Law of Cosines, we can then figure out the angle between two vectors. Given two vectors $$v$$ and $$u$$ that $$\textit{span}$$ a plane in $$\mathbb{R}^{n}$$, we can then connect the ends of $$v$$ and $$u$$ with the vector $$v-u$$.

Then the Law of Cosines states that:

$\|v-u\|^2 = \|u\|^2 + \|v\|^2 - 2\|u\|\, \|v\| \cos \theta$

Then isolate $$\cos \theta$$:

\begin{eqnarray*}
\|v-u\|^{2} - \|u\|^{2} - \|v\|^{2} &=& (v^{1}-u^{1})^{2} + \cdots + (v^{n}-u^{n})^{2} \\
& & \quad - \big((u^{1})^{2} + \cdots + (u^{n})^{2}\big) \\
& & \quad - \big((v^{1})^{2} + \cdots + (v^{n})^{2}\big) \\
& = & -2 u^{1}v^{1} - \cdots - 2u^{n}v^{n}
\end{eqnarray*}
Thus,

$\|u\|\, \|v\| \cos \theta = u^{1}v^{1} + \cdots + u^{n}v^{n}\, .$

Note that in the above discussion, we have assumed (correctly) that Euclidean lengths in $$\mathbb{R}^{n}$$ give the usual notion of lengths of vectors for any plane in $$\mathbb{R}^{n}$$. This now motivates the definition of the dot product.

Definition

The $$\textit{dot product}$$ of two vectors $$u=\begin{pmatrix}u^{1} \\ \vdots \\ u^{n}\end{pmatrix}$$ and $$v=\begin{pmatrix}v^{1} \\ \vdots \\ v^{n}\end{pmatrix}$$ is

$u\cdot v := u^{1}v^{1} + \cdots + u^{n}v^{n}\, .$

The $$\textit{length}$$ or $$\textit{norm}$$ or $$\textit{magnitude}$$ of a vector is

$\|v\| := \sqrt{v\cdot v }\, .$

The $$\textit{angle}$$ $$\theta$$ between two vectors is determined by the formula $$u\cdot v = \|u\|\|v\|\cos \theta\, .$$

Remark

When the dot product between two vectors vanishes, we say that they are perpendicular or $$\textit{orthogonal}$$. Notice that the zero vector is orthogonal to every vector.

The dot product has some important properties:

1. The dot product is $$\textit{symmetric}$$, so $$u\cdot v = v\cdot u\, ,$$
2. $$\textit{Distributive}$$ so $$u\cdot (v+w) = u\cdot v + u\cdot w\, ,$$
3. $$\textit{Bilinear}$$, which is to say, linear in both $$u$$ and $$v$$. Thus $$u\cdot (cv+dw) = c \, u\cdot v +d \, u\cdot w\, ,$$ and $$(cu+dw)\cdot v = c\, u\cdot v + d\, w\cdot v\, .$$
4. $$\textit{Positive Definite}$$: $$u\cdot u \geq 0\, ,$$ and  $$u\cdot u = 0$$ only when $$u$$ itself is the $$0$$-vector.

There are, in fact, many different useful ways to define lengths of vectors. Notice in the definition above that we first defined the dot product, and then defined everything else in terms of the dot product.  So if we change our idea of the dot product, we change our notion of length and angle as well.  The dot product determines the $$\textit{Euclidean length and angle}$$ between two vectors.

Other definitions of length and angle arise from $$\textit{inner products}$$, which have all of the properties listed above (except that in some contexts the positive definite requirement is relaxed). Instead of writing $\cdot$ for other inner products, we usually write $$\langle u,v \rangle$$ to avoid confusion.

Example 46

Consider a four-dimensional space, with a special direction which we will call "time''. The $$\textit{Lorentzian inner product}$$ on $$\mathbb{R}^{4}$$ is given by $$\langle u,v \rangle = u^{1}v^{1} + u^{2}v^{2} + u^{3}v^{3} - u^{4}v^{4}$$.  This is of central importance in Einstein's theory of special relativity. Note, in particular,  that it is not positive definite.

As a result, the "squared-length'' of a vector with coordinates $$x, y, z$$ and $$t$$ is $$\|v\|^{2} = x^{2} + y^{2} + z^{2} - t^{2}$$. Notice that it is possible for $$\|v\|^{2}\leq 0$$ even with non-vanishing $$v$$!

Theorem    Cauchy-Schwarz Inequality

For non-zero vectors $$u$$ and $$v$$ with an inner-product $$\langle\:\ ,\:\, \rangle$$,

$\frac{|\langle u,v \rangle|}{\|u\|\, \|v\|} \leq 1$

Proof

The easiest proof would use the definition of the angle between two vectors and the fact that $$\cos \theta \leq 1$$. However, strictly speaking we did not check our assumption that we could apply the Law of Cosines to the Euclidean length in $$\mathbb{R}^{n}$$. There is, however a simple algebraic proof. Let $$\alpha$$ be any real number and consider the following positive, quadratic polynomial in $$\alpha$$

$0\leq \langle u+\alpha v,u+\alpha v\rangle = \langle u,u\rangle +2\alpha \langle u,v\rangle +\alpha^{2} \langle v,v\rangle\, .$

You should carefully check for yourself exactly which properties of an inner product were used to write down the above inequality!

Next, a tiny calculus computation shows that any quadratic $$a\alpha^{2} + 2b \alpha + c$$ takes its minimal value $$c-\frac{b^{2}}{a}$$
when $$\alpha=-\frac{b}{a}$$. Applying this to the above quadratic gives

$0\leq \langle u,u\rangle -\frac{\langle u,v\rangle^{2}}{\langle v,v\rangle}\, .$

Now it is easy to rearrange this inequality to reach the Cauchy--Schwarz one above.

Theorem    Triangle Inequality

Given vectors $$u$$ and $$v$$, we have:
$\|u+v\| \leq \|u\| + \|v\|$

Proof

\begin{eqnarray*}
\|u+v\|^{2} & = & (u+v)\cdot (u+v) \\
& = & u\cdot u + 2 u\cdot v + v\cdot v \\
& = & \|u\|^{2} + \|v\|^{2} + 2\,  \|u\|\,  \|v\| \cos \theta \\
& = & \left(\|u\| + \|v\|\right)^{2} + 2 \, \|u\| \, \|v\| (\cos \theta -1) \\
& \leq & \left(\|u\| + \|v\|\right)^{2}    \\
\end{eqnarray*}

Then the square of the left-hand side of the triangle inequality is $$\leq$$ the right-hand side, and both sides are positive, so the result is true.

The triangle inequality is also "self-evident'' examining a sketch of $$u$$, $$v$$ and $$u+v$$:

Example 47

Let

$$a=\begin{pmatrix}1\\2\\3\\4\end{pmatrix} \mbox{ and }\ b = \begin{pmatrix}4\\3\\2\\1\end{pmatrix}\, ,$$

so that

$$a\cdot a= b\cdot b =1 +2^{2}+3^{2}+4^{2}=30$$ $$\Rightarrow \|a\|=\sqrt{30} =\| b \| \mbox{ and } \ \big(\|a\|+\|b\|\big)^{2}=(2\sqrt{30})^{2}=120\, .$$

Since

$$a+b= \begin{pmatrix}5\\5\\5\\5\end{pmatrix}\, , \quad$$

we have

$$\|a+b\|^{2}=5^{2}+5^{2}+5^{2}+5^{2}=100<120=\big(\|a\|+\|b\|\big)^{2}$$

as predicted by the triangle inequality.

Notice also that $$a\cdot b=1.4+2.3+3.2+4.1=20< \sqrt{30}.\sqrt{30}=30=\|a\|\, \|b\|$$ in accordance with the Cauchy--Schwarz inequality.


text:
Conference PaperPDF Available

# Towards Improving Privacy of Synthetic DataSet

Authors:

## Abstract and Figures

Recent growth in domain specific applications of machine learning can be attributed to availability of realistic public datasets. Real world datasets may always contain sensitive information about the users, which makes it hard to share freely with other stake holders, and researchers due to regulatory and compliance requirements. Synthesising datasets from real data by leveraging generative techniques is gaining popularity. However, the privacy analysis of these dataset is still a open research. In this work, we fill this gap by investigating the privacy issues of the generated data sets from attacker and auditor point of view. We propose instance level Privacy Score (PS) for each synthetic sample by measuring the memorisation coefficient \textit{$\boldsymbol{\alpha_{m}}$} per sample. Leveraging, PS we empirically show that accuracy of membership inference attacks on synthetic data drop significantly. PS is a model agnostic, post training measure, which helps data sharer with guidance about the privacy properties of a given sample but also helps third party data auditors to run privacy checks without sharing model internals. We tested our method on two real world data sets and show that attack accuracy reduced by PS based filtering.
Content may be subject to copyright.
Towards Improving Privacy of Synthetic
DataSets
1UCD School of Computing, Dublin Ireland
2Tenable Network Security
an.lekhac@@ucd.ie
Abstract. Recent growth in domain speciﬁc applications of machine
learning can be attributed to availability of realistic public datasets. Real
world datasets may always contain sensitive information about the users,
which makes it hard to share freely with other stake holders, and re-
searchers due to regulatory and compliance requirements. Synthesising
datasets from real data by leveraging generative techniques is gaining
popularity. However, the privacy analysis of these dataset is still a open
research. In this work, we ﬁll this gap by investigating the privacy issues
of the generated data sets from attacker and auditor point of view. We
propose instance level Privacy Score (PS) for each synthetic sample by
measuring the memorisation coeﬃcient αmper sample. Leveraging, PS
we empirically show that accuracy of membership inference attacks on
synthetic data drop signiﬁcantly. PS is a model agnostic, post training
measure, which helps data sharer with guidance about the privacy prop-
erties of a given sample but also helps third party data auditors to run
privacy checks without sharing model internals. We tested our method
on two real world data sets and show that attack accuracy reduced by
PS based ﬁltering.
Keywords: Privacy Preserving Synthetic Data ·Generative Adversarial
Networks ·Privacy Audit.
1 Introduction
Machine learning (ML) has been widely adopted in a variety of applications
across multiple domains. Due to the maturity of large-scale training infrastruc-
ture [3], development tools and, the availability of large amounts of training data
companies are building huge models with billions of parameters for commercial
applications. The training data used to train these models often contain per-
sonal or sensitive data [1], which was aggregated from datasets that are used
for purposes unknown to the data owner (i.e. the person itself). Unfortunately,
end-users who play an active role as data donors lack visibility into how this data
is being used/misused by the applications built on top of the data collected. A
2 Kuppa et al.
growing number of AI incidents3show an upward trend of data and model mis-
uses, which include discriminatory hiring decisions, behavioral manipulations,
and unethical systems that lack transparency. To gain the trust of users and
avoid the privacy implications of data sharing for end users institutions [4–6]
and companies [9–16] are adopting synthetic data [17] as a solution to avoid
privacy issues.
Synthetic data as an idea was ﬁrst adopted in designing various imputation
methods in the early 90’s [17, 18]. Raghunathan et.al [32] formally deﬁned a
framework to deﬁne properties of synthetic data, which was further enhanced in
multiple other researchers [19–22,33,34]. Many recent works leveraged traditional
statistical techniques and machine learning models to generate synthetic data
[23, 25, 26, 28–31].
Ideal synthetic data has properties such as handling large datasets with di-
verse data types (categorical, ordinal, continuous, skewed, etc.), preserving se-
mantic relationships between variables [26] and, underlying algorithms are fairly
generalizable and needs a little tuning. ML models trained on synthetic data
may have similar performance as original data. Theoretical privacy guarantees
can be achieved via ”ideal” synthetic data as a synthetic data record is not an
actual data record of an individual but reﬂect the exact data distributions of
original data.
In the context of data sharing, the majority of recent methods employ gen-
erative models [39, 40, 47, 50, 51] to obtain the synthetic data from real data.
Popular generative modeling frameworks such as Generative Adversarial Net-
works (GANs [40]) and Variational Auto-encoders (VAEs) use real data during
training to train a model capable of generating (or ”synthesising”) ”fake” data
points. Quality metrics such as ﬁdelity which measures how “realistic” is the
synthetic data when compared to real data and, diversity which measures how
diverse the generated samples to cover the variability of real data are used to
assess quality of the synthetic data.
Measuring the generalization property of synthetic data is very important
and often ignored by synthetic data generators. Most of the proposed methods
optimize on ﬁdelity and diversity properties ignoring to measure or report, how
well the synthetic samples capture the overall distribution including outliers of
real data and, not just copies of the (real) samples of training data. Evaluat-
ing generalization in generative models is not easy [54, 55] when compared to
discriminative models which are tested on held-out data.
From compliance and regulation view point, the General Data Protection
Regulation (GDPR [42]) gives new rights to citizens regarding transparency
around personal data. These include the right of access (knowing which data
about the person has been stored) and the right to data erasure, also known
as the right to be forgotten (having personal data deleted under certain cir-
cumstances). Privacy risk include the risk of re-identiﬁcation and the risk of
inference [35–37, 41]. When an attacker is able to identify and link individuals
within a dataset or across multiple datasets, then these datasets pose the risk
3https://incidentdatabase.ai/
Towards Improving Privacy of Synthetic DataSets 3
of re-identiﬁcation. These types of attacks can reveal the presence of user [43] or
user data was used to train an ML model [44]. Deducing the private information
of individuals with high conﬁdence from the linked datasets or ML model fall
under the risk of inference.
Motivated by the need from compliance, privacy of the end user and, grow-
ing popularity of synthetic data as sharing mechanism both in industry and
academia, in this work we aim to investigate privacy properties of synthetic data
generated by generative models. First, we propose a novel Membership Inference
Attack (MIA), which uses the popular bagging technique to achieve attacker goal
of identifying the membership of user in the model trained on synthetic data.
Next, a instance level privacy score is calculated to ﬁlter the synthetic samples,
which could compromise the privacy of the sensitive attributes in the synthetic
data to protect from MIAs.
2 Background
We present a brief background on membership inference attacks and generative
machine learning models.
Membership inference attack (MIA). is one of the most popular privacy
attacks against ML models [73–79]. The attacker goal of membership inference
(MI) is to determine whether a data sample xis part of the training dataset of
a target model T. We deﬁne a membership inference attack model AMemInf as
a binary classiﬁer.
AMemInf :x, T 7→ {member,non-member}(1)
MIA poses severe privacy risks, using a model trained on data samples of
people with sensitive information an attacker can execute MIA to successfully
infer the membership of person revealing the person’s sensitive information. To
achieve MIA, Shokri et al [73] train a number of shadow classiﬁers on conﬁdence
scores produced by the target model with labels indicating whether samples
came from the training or testing set. MIAs are shown to work in white-box [57]
as well as black-box scenarios against a various target models including ML-as-
a-service [73] and generative models [56]. Yeom et al. [59] explore overﬁtting as
the root cause of MI vulnerability.
Similarly, a variant of MIA is attribute inference in which sensitive attribute
can be inferred from the target model. This sensitive attribute is not related to
the target ML model’s original classiﬁcation task. For instance, a target model
is designed to classify an individual’s age from their social network posts, while
attribute inference aims to infer their educational background [71, 72]. Some
studies [58,59, 64] attribute membership inference attacks to the generalization
gap, the over-ﬁtting of the model, and data memorization capabilities of neu-
ral networks. Deep neural networks have been shown to memorize the training
data [63, 69, 70], rather than learning the latent properties of the data, which
means they often tend to over-ﬁt to the training data. Weakness like this, in
4 Kuppa et al.
some cases, can be exploited by an adversary to infer data samples’ sensitive
attributes.
Given a data sample xand its representation from a target model, denoted
by h=f(x), to conduct the attribute inference attack, the adversary trains an
attack model AAttInf formally deﬁned as follows:
AAttInf :h7→ s(2)
where srepresents the sensitive attribute.
ing system, where two competing models are trained against each other. They
comprise of two components, a discriminator (D), which is trained to diﬀerenti-
ate between real generated data; and the generator (G), which creates fake data
to fool the discriminator. The generator Gtakes in a random noise vector Zand
parameter vector thetagand learns to generate synthetic samples very similar
to the real data. The resulting G(Z;θg) network is is used to generate fake data
with data distribution Fg.
Training GAN’s on real-world datasets is notoriously hard. One has to care-
fully tune both generator and discriminator network parameters at training time
to avoid conditions like mode collapse and vanishing gradients. Both these con-
ditions aﬀect the quality of synthetic data generated. Choice of loss functions
and GAN architectures also play a vital role in avoiding these problems [65].
Tabular Generators. In this work, we deal in the feature space, which is tab-
ular in nature. The values of features are continuous, ordinal, categorical, tex-
tual, and/or multi-categorical values. Textual values can be of variable length,
high-dimensional due to long-tail distributions of observational data. Textual
data is converted into a set of sequences and projected into a trained embed-
ding that preserves semantic meaning similar to methods proposed in Natural
work (GAN) [40] architectures proposed in literature perform poorly on tabular
data [66] due to – (a) Real data often consists of mixed data types. To generate
this data simultaneously, a GAN must be able to apply both softmax and tanh
on the output; (b) Images pixel values generally follow a Gaussian-like distribu-
tion, which can be normalized to [-1,1], whereas with tabular data, continuous
data often have non-Gaussian distributions with long tails that leads to gradi-
ent saturation; continuous data type and imbalanced categorical columns need
preprocessing steps before feeding into generators.
Recently proposed Conditional Tabular GAN or CTGAN [66] addresses some
of the key problems for tabular data synthesis. The training process involves
simulating records one by one. It ﬁrst randomly selects one of the variables, then
it randomly selects a value for that variable. Given that value for that variable,
the algorithm ﬁnds a matching row from the training data. It also generates the
rest of the variables conditioning on the selected variable. The generated and true
rows are sent to the critic that gives a score. Similarly, modiﬁed WGAN-GP [68]
changed the generator of the WGAN-GP. Figure 1 summarizes the CTGAN and
Towards Improving Privacy of Synthetic DataSets 5
WGAN-GP training procedures. to handle continuous variables. Other recently
proposed Robust Variational AutoEncoders (R-VAEs) [67], which inherits the
properties of typical Variational AutoEncoders, handles the tabular data more
eﬃciently.
Randomly
select a value
Randomly
select a
column
Noise Z
Fz
N(0,Ik)
Generate record
with value
v G(Z;θg|v)
Fake record
with v
Randomly
select record
in Xwith v
Real record
with v
Critic
D(·)
Data X
Score
(a) CTGAN
Noise Z
Fz
N(0,Ik)
Generator
G(Z;θg)Fake Data Fg
Real Data FX
Critic
D(·)
D: set
of 1-
Lipschitz
functions
Data X
Score of
realism Loss
maxD∈D{D(X)} − [D{G(Z;θg)}]
max
θg
[D{G(Z;θg)}]
(b) WGAN-GP
Fig. 1: CTGAN and WGAN-GP Training dynamics
DPGAN [52] and PATEGAN [61] are popular diﬀerentially private GAN’s.
DPGAN adds noise to the gradient of the discriminator during training to create
diﬀerential privacy guarantees. PATEGAN ﬁrst trains multiple teacher models
on disjoint distributions of data. At inference time, to classify a new sample it
noisily aggerates outputs from previously trained teacher models to generate the
ﬁnal output.
3 Instance level Privacy Scoring
To avoid MIA type attacks and unintentional leakage of sensitive attributes, we
devise an instance level privacy score for generated synthetic data. As shown
in previous studies, the main reasons for privacy compromises in generative
models is mainly because of over ﬁtting of training data that could lead to
data memorization. We empirically measure memorization coeﬃcient αmfor
each generated sample and use this measure as privacy score. When distributing
synthetic data to external third parties, users can ﬁlter samples with high privacy
score and reduce privacy risks.
Given Xr,Xsreal and synthetic data set generated by some generative al-
gorithm Φrespectively. We ﬁrst convert into latent vectors in embedding space
E(Xr), E(Xs). Next, we measure the spherelet distance between each real and
generated sample latent vectors. The spherelet distance between s(Xi) and s(Xj)
is deﬁned as:
dS=dR(Xi, Xj) + dE(Xi, Xj) (3)
where dEis the Euclidean distance between two sets of points, and dRis Rieman-
nian divergence (geometric) of dataset of Xiand Xj. The intuition behind this is
6 Kuppa et al.
we are measuring the distance between points by projecting the samples on to a
sphere centered at cwith radius r. Spherelet of Xdenoted by s(X) = S(V, c, r)
where Vdetermines an aﬃne subspace the sphere lies in. The spherical error [60]
of Xis deﬁned by
(X) = 1
n
n
X
i=1
(kxick − r)2(4)
If xilie on the sphere then (X) = 0. Also the distance metric captures (a)
which points are found in data manifold in low probability; (b) how close the
synthetic and real in nearest neighbour proximity; which can give us a proxy
measure for over-ﬁtting and data memorization. memorization coeﬃcient αmfor
a generated sample Xgiis deﬁned by gni/tniwhere tniand gniare sample counts
from training and generated dataset, which have proximity distance lesser than
a threshold > λ. In short, we are measuring the support set for the sample Xgiin
the training and synthetic set. Samples, which have αm<1 have large inﬂuence
of training set and are prone to MI attacks vs samples with αm>1 capture
the underlying distribution of training data without leaking private training set
attributes.
The advantage of this method is the model Φis agnostic and works on data
level. Users can discard the samples which cross a certain threshold privacy
score to protect privacy of users and, data audit by compliance bodies can be
performed without a need to know the model internals and training dynamics.
4 MIA Method
Membership Inference The goal of a membership inference attack is to create a
function that accurately predicts whether a data point belongs to the training set
of the classiﬁer. To achieve this, an attacker trains a set of surrogate models [73]
on conﬁdence scores produced by the target model and membership as labels to
achieve her goal.
Typically, the attacker exploits the model conﬁdence reﬂecting overﬁtting to
the conﬁdence scores for each input data sample. Their approach involved ﬁrst
training many ’surrogate’ ML models similar to the target model and uses these
shadow models to train a membership inference model to identify whether a
sample was in the training set of any of the shadow models. MIA is evaluated as
a binary classiﬁcation task, where metrics like false-negative, false-positive rates,
and F1 scores reﬂect the attack success rates of predicting that a given sample
is part of the training set.
We adapt ’shadow’ training model in our MI attack on synthetic data. Given
an auxiliary dataset Daux similar to the training data distribution, we ﬁrst ran-
domly sample single data point Ltand remove it from the dataset. Now, a
well-known ensemble learning method Bootstrap aggregating (Bagging) is used
to divide remaining dataset Daux 1 into subsamples with replacement. Gener-
ative models are trained on each sub sample and using the trained models syn0
Towards Improving Privacy of Synthetic DataSets 7
synthetic datasets of size kis sampled. These samples as labeled lj= 0 indicat-
ing that synthetic data does not contain target sample Lt. Same procedure is
repeated by adding Ltinto the dataset Daux and the synthetic data sets syn1of
size kis sampled. These samples as labled lj= 1 indicating that synthetic data
contain target sample Lt. Now we have labeled dataset that captures the mem-
bership relationship and we train a random forest classiﬁer MIc, which predicts
the probability of a given sample is member or not of the original data set. Given
a synthetic dataset SAux,M Icpredicts the membership of the target record tin
the training set of the generative model Gthat produced SAux .
5 Experiments
The membership privacy leakage is evaluated through a membership privacy
experiment [48,49]. Given a training set Dtwhich follows a data distribution D
and a learning algorithm φis trained on Dt. We assume the adversary has only
access to φvia a query interface, which exposes inference and class probability
of a given sample. In synthetic data sharing use cases, the adversary has access
to synthetic datasets generated by some generative model. Here the goal of the
adversary is to discover the membership of a given sample i.e. it belongs to
training data or not with access to only synthetic data. We assume domain-
strength of the proposed Instance Level Privacy ﬁltering of synthetic data we
measure the accuracy drop of MI attack with/without ﬁltering mechanism.
For this work, we use two datasets – First dataset is SIVEP-Gripe database [2]
released by Brazilian government with records of 99,557 ICU Covid-19 patients
in Brazil, which includes personal information (ethnicity, age and location). A
logistic regression (binary classiﬁcation) model is ﬁt to predict patient-level
COVID-19 mortality with 6882 data sample and 25 features. Second dataset
is the Brazilian E-Commerce Public Dataset by Olist4. It consists of 112,000
customer orders at multiple marketplaces in Brazil. The dataset contains order
status, price, payment and freight performance to customer location, product
attributes and ﬁnally reviews written by customers. Customer and Seller Id’s
are anonymized, and references to the companies and partners in the review
text have been replaced with the names of Game of Thrones great houses. For
our experiment we use columns price, customer identiﬁer, customer zip code,
customer city as feature columns. A logistic regression model is ﬁt to predict the
customer spend by aggregating price column.
We create 5 synthetic datasets using CTGAN, WGAN-GP, PATEGAN, DP-
GAN, and, RVAE as synthetic data generators for SIVEP-Gripe and E-Commerce
data. The training of generator the input dimensions of architecture is ad-
justed to accommodate the dataset shape. All hidden layers use the ReLU
activation function, the ﬁnal layer uses the sigmoid function. The discrimi-
nator network also uses three hidden layers, which are symmetric to the gen-
erator. The hidden layers again use ReLU. We train each network with 100
4https://www.kaggle.com/olistbr/brazilian-ecommerce/
8 Kuppa et al.
(a)
(b)
Fig. 2: Synthetic data size dependency on MIA attack for (a) E-Commerce
Dataset (b) EHR dataset
Towards Improving Privacy of Synthetic DataSets 9
(a)
(b)
Fig. 3: Attack Accuracy drop before and after ﬁltering data points with high
privacy scores - (a) EHR dataset (b) E-Commerce Dataset
10 Kuppa et al.
epochs. For training diﬀerential private GAN’s, we used the privacy budgets
of = [0.01,0.1,0.5,1.0,3.0]. λis set to 0.5 to reﬂect the adversary advantage
in membership attack. Randomly 1000 samples are sampled from training and
marked as Daux. To reﬂect maximum compromise by adversary as white box
setting, we use the same generator to train the shadow model S.
5.1 Experimental Results and Discussion
We ﬁrst present experimental results of the membership inference attack and the
attack accuracy drop before and after we ﬁlter the data with αmmethod. We
measure the accuracy of attack with respect to number of samples in the auxiliary
dataset used in the attack mechanism. Figure 5.1 and 3 summarizes the privacy
risk of each synthesising method in terms of MI attack accuracy and auxiliary
dataset. We observe that increasing the size of the synthetic dataset increases the
the privacy of the data generator weakens. Similarly, diﬀerential privacy methods
that add noise to the output with some privacy budget , the attack accuracy
decreases showing the value of DP-based generators reduce privacy risks. Increase
in privacy strength of the dataset is consistent across datasets and diﬀerent
models highlighting the strength of our proposed method.
5.2 Discussion
To develop good synthetic data generation models we still have some technical
hurdles, for example, it is not only hard to exactly predict what data charac-
teristics will be preserved in a model’s stochastic output but also appropriate
metrics to empirically compare two distributions are lacking. The privacy bud-
gets in terms of perturbations to achieve diﬀerential privacy make it even harder
to predict which records will remain vulnerable, and might even increase the
exposure of some data records. Our work aims to give some direction towards
improving synthetic data privacy by assigning instance-level privacy scores. A
data auditor or service provider can leverage the scores to remove the data
points prone to MI attacks. The advantage of our method is a model agnostic
and it can be used in post-hoc fashion. We explored the accuracy drop from
the attacker end, but measuring the accuracy-privacy tradeoﬀ due to dropping
the data points from the synthetic data on downstream models needs further
investigation.
We want to highlight some of the limitations of our work. We only tested
our method on tabular data, further investigation is needed to understand the
privacy risks of other types of data such as text, images and, audio. Additionally,
it can be observed that we use the whole dataset to measure αm, which may
be misleading as pointed out by [46], membership leakage for subgroup may
be more vulnerable than others for discriminative models. We plan to address
this drawback in our future work. It is also important to point out that the
privacy risk measures provided in this analysis are dataset-dependent and can
be generator-speciﬁc. While the proposed method gives some empirical privacy
Towards Improving Privacy of Synthetic DataSets 11
guarantees for generative models and guidance for data sharing use case, we need
more detailed analysis on the dependency of αmagainst model parameters, and
varied dataset distributions. In terms of threat models, our analysis does not
take into consideration an attacker who has knowledge of the ﬁltering scheme in
advance. A malicious attacker can use only ﬁltered data points to train surrogate
models making the proposed scheme invalid.
6 Conclusion
One of the major impediments in advancing ML in security-sensitive domains is
the lack of open realistic publicly available datasets. To gain trust and protect the
end-user data and, comply with regulations stakeholders are leveraging synthetic
data as a tool to share data between third-party aggregators and researchers.
However, the privacy analysis of these datasets is still open research. In this
work, we ﬁll this gap by investigating the privacy issues of the generated data
sets from the attacker and auditor point of view. We propose an Instance-level
Privacy Score (PS) for each synthetic sample by measuring the memorization
coeﬃcient αmper sample. Leveraging, PS we empirically show that the accu-
racy of membership inference attacks on synthetic data drops signiﬁcantly. PS is
a model agnostic, post-training measure, which helps data sharer with guidance
about the privacy properties of a given sample but also helps third party data
auditors to run privacy checks without sharing model internals. Finally, there
is a lot more work to be done purely within the realm of privacy and our work
only addresses a very small issue empirically. Understanding social and legal im-
plications of synthetic data sharing needs inputs and partnerships from policy
experts and, compliance oﬃcers.
References
1. Carlini, Nicholas and Tramer, Florian and Wallace, Eric and Jagielski, Matthew
and Herbert-Voss, Ariel and Lee, Katherine and Roberts, Adam and Brown, Tom
and Song, Dawn and Erlingsson, Ulfar and others
Extracting Training Data from Large Language Models , arXiv preprint
arXiv:2012.07805
Ministry of Health. SIVEP-Gripe public dataset, (accessed May 10, 2020; in Por-
tuguese), 2020.
3. N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates,
S. Bhatia, N. Boden, A. Borchers, et al. In-datacenter performance analysis of a
tensor processing unit. In Proceedings of the 44th Annual International Symposium
on Computer Architecture, pages 1–12, 2017.
4. Departement of Commerce, National Institute of Standards and Technology, “Diﬀer-
ential private synthetic data challenge.” https://www.challenge.gov/challenge/
differential-privacy-synthetic-data- challenge/, 2019. Accessed 2021-02-19.
5. T. Theraux, Olivier, “Anonymisation and synthetic data:
towards trustworthy data.” https://theodi.org/article/
12 Kuppa et al.
anonymisation-and-synthetic-data- towards-trustworthy-data/, 2019. Ac-
cessed 2021-02-19.
6. The Open Data Institute, “Diagnosing the nhs: Synae.” https://www.odileeds.
org/events/synae/. Accessed 2021-02-19.
7. Rubin, D. B. Discussion: Statistical Disclosure Limitation, 1993.
8. Rubin, D. B. Multiple imputation after 18+ years. American Statistical Association,
91(434):473–489, 1996.
9. “Hazy.” https://hazy.com/.
10. “AIreverie.” https://aireverie.com/.
11. “Statice.” https://statice.ai/.
12. “one-view.” https://one-view.ai/.
13. “datagen.” https://www.datagen.tech/.
14. “Synthesize.” https://synthezise.io/.
15. “cognata.” https://www.cognata.com/.
16. “Mostly-ai.” https://mostly.ai/.
17. D. B. Rubin, “Statistical disclosure limitation,” Journal of oﬃcial Statistics, vol. 9,
no. 2, pp. 461–468, 1993.
18. Little, R. J. Statistical Analysis of Masked Data. Journal of Oﬃcial Statistics, 9
(2):407–426, 1993.
19. Abowd, J. M. and Lane, J. New Approaches to Conﬁdentiality Protection: Syn-
thetic Data, Remote Access and Research Data Centers. In Domingo-Ferrer, J. and
Torra, V. (eds.), Privacy in Statistical Databases, volume 3050, pp. 282–289. 2004.
ISBN 978-3-540-22118-0.
20. Abowd, J. M. and Woodcock, S. D. Multiply-Imputing Conﬁdential Characteristics
and File Links in Longitudinal Linked Data. In Privacy in Statistical Databases,
volume 3050, pp. 290–297. 2004. ISBN 3-540-22118-2.
21. Reiter, J. P. and Raghunathan, T. E. The multiple adaptations of multiple impu-
tation. Journal of the American Statistical Association, 102(480):1462–1471, 2007.
22. Drechsler, J. and Reiter, J. P. Sampling with synthesis: A new approach for releas-
ing public use census microdata. Journal of the American Statistical Association,
105(492):1347–1357, 2010.
23. Drechsler, J. and Reiter, J. P. An Empirical Evaluation of Easily Implemented,
Nonparametric Methods for Generating Synthetic Datasets. Computational Statis-
tics and Data Analysis, 55(12):3232–3243, dec 2011.
24. Kinney, S. K., Reiter, J. P., Reznek, A. P., Miranda, J., Jarmin, R. S., and Abowd,
J. M. Towards Unrestricted Public Use Business Microdata: The Synthetic Longi-
tudinal Business Database. International Statistical Review, 79(3):362–384, 2011.
25. Reiter, J. P. Using cart to generate partially synthetic, public use microdata.
Journal of Oﬃcial Statistics, 21(3):441–462, 2005a.
26. Caiola, G. and Reiter, J. P. Random Forests for Generating Partially Synthetic,
Categorical Data. Transactions on Data Privacy, 3(1):27–42, 2010.
27. Drechsler, J. Synthetic Datasets for Statistical Disclosure Control, volume 53.
Springer, 2011. ISBN 9788578110796.
28. Bowen, C. M. and Liu, F. Comparative Study of Diﬀerentially Private Data Syn-
thesis Methods. Statistical Science, forthcoming.
29. Drechsler, J. Synthetic Datasets for Statistical Disclosure Control, volume 53.
Springer, 2011. ISBN 9788578110796.
30. Manrique-Vallier, D. and Hu, J. Bayesian Non-Parametric Generation of Fully
Synthetic Multivariate Categorical Data in the Presence of Structural Zeros. Journal
of the Royal Statistical Society. Series A: Statistics in Society, 181(3):635–647, 2018.
Towards Improving Privacy of Synthetic DataSets 13
31. Snoke, J., Raab, G., Nowok, B., Dibben, C., and Slavkovic, A. General and speciﬁc
utility measures for synthetic data. 2016.
32. Raghunathan, T. E., Reiter, J. P., and Rubin, D. B. Multiple imputation for
statistical disclosure limitation. Journal of oﬃcial statistics, 19(1):1, 2003.
33. Kinney, S. K., Reiter, J. P., Reznek, A. P., Miranda, J., Jarmin, R. S., and Abowd,
J. M. Towards Unrestricted Public Use Business Microdata: The Synthetic Longi-
tudinal Business Database. International Statistical Review, 79(3):362–384, 2011.
34. Kinney, S. K., Reiter, J. P., and Berger, J. O. Model Selection When Multiple
Imputation is Used to Protect Conﬁdentiality in Public Use Data. Journal of Privacy
and Conﬁdentiality, 2(2):3–19, 2010.
35. Article 29 Data Protection Working Party - European Commission, “Opin-
ion 05/2014 on anonymisation techniques.” https://ec.europa.eu/justice/
article-29/documentation/opinion-recommendation/files/2014/wp216_en.
pdf, 2014.
36. M. Elliot, E. Mackey, K. O’Hara, and C. Tudor, The anonymisation decision-
making framework. UKAN Manchester, 2016.
37. I. S. Rubinstein and W. Hartzog, “Anonymization and risk,” Wash. L. Rev., vol. 91,
p. 703, 2016.
38. M. Elliot, K. O’hara, C. Raab, C. M. O’Keefe, E. Mackey, C. Dibben, H. Gowans,
K. Purdam, and K. McCullagh, “Functional anonymisation: Personal data and the
data environment,” Computer Law & Security Review, vol. 34, no. 2, pp. 204–221,
2018.
39. N. Patki, R. Wedge, and K. Veeramachaneni, “The synthetic data vault,” in 2016
IEEE International Conference on Data Science and Advanced Analytics (DSAA),
pp. 399–410, IEEE, 2016.
40. I. Goodfellow, “Nips 2016 tutorial: Generative adversarial networks,” arXiv
preprint arXiv:1701.00160, 2016.
41. M. Elliot, K. O’hara, C. Raab, C. M. O’Keefe, E. Mackey, C. Dibben, H. Gowans,
K. Purdam, and K. McCullagh, “Functional anonymisation: Personal data and the
data environment,” Computer Law & Security Review, vol. 34, no. 2, pp. 204–221,
2018.
42. European Commission (2016). Regulation (EU) 2016/679: General Data Protection
Regulation (GDPR).
43. L. Sweeney, “k-anonymity: A model for protecting privacy,” International Journal
of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 10, no. 05, pp. 557–
570, 2002.
44. R. Shokri, M. Stronati, C. Song, and V. Shmatikov, “Membership Inference Attacks
Against Machine Learning Models,” in IEEE Symposium on Security and Privacy
(S&P), 2017.
45. A. Machanavajjhala, J. Gehrke, D. Kifer, and M. Venkitasubramaniam, “L-
diversity: privacy beyond k-anonymity,” in 22nd International Conference on Data
Engineering (ICDE’06), 2006.
46. M. Yaghini, B. Kulynych, and C. Troncoso, “Disparate vulnerability: On the unfair-
ness of privacy attacks against machine learning,” arXiv preprint arXiv:1906.00389,
2019.
47. H. Ping, J. Stoyanovich, and B. Howe, “Datasynthesizer: Privacy-preserving syn-
thetic datasets,” in Proceedings of the 29th International Conference on Scientiﬁc
and Statistical Database Management, 2017.
48. B. Jayaraman, L. Wang, D. Evans, and Q. Gu, “Revisiting membership inference
under realistic assumptions,” arXiv preprint arXiv:2005.10881, 2020.
14 Kuppa et al.
49. S. Yeom, I. Giacomelli, M. Fredrikson, and S. Jha, “Privacy risk in machine learn-
ing: Analyzing the connection to overﬁtting,” in 2018 IEEE 31st Computer Security
Foundations Symposium (CSF), pp. 268–282, IEEE, 2018.
50. J. Zhang, G. Cormode, C. M. Procopiuc, D. Srivastava, and X. Xiao, “Privbayes:
Private data release via bayesian networks,” ACM Trans. Database Syst., 2017.
51. L. Xu, M. Skoularidou, A. Cuesta-Infante, and K. Veeramachaneni, “Modeling
tabular data using conditional gan,” in Advances in Neural Information Processing
Systems, 2019.
52. Yoon, J., Drumright, L. N., and Van Der Schaar, M. Anonymization through
Biomedical and Health Informatics, 2020.
53. Brock, A., Donahue, J., and Simonyan, K. Large scale gan training for high ﬁdelity
natural image synthesis. arXiv preprint arXiv:1809.11096, 2018.
54. Adlam, B., Weill, C., and Kapoor, A. Investigating under and overﬁtting in wasser-
stein generative adversarial networks. arXiv preprint arXiv:1910.14137, 2019.
55. Meehan, C., Chaudhuri, K., and Dasgupta, S. A non-parametric test to detect
data-copying in generative models. arXiv preprint arXiv:2004.05675, 2020.
56. Jamie Hayes, Luca Melis, George Danezis, and Emiliano De Cristofaro. Logan:
Membership inference attacks against generative models. Proceedings on Privacy
Enhancing Technologies, 2019(1):133–152, 2019.
57. Sablayrolles, A.; Douze, M.; Ollivier, Y.; Schmid, C.; and J´egou, H. 2019. White-
box vs Black-box: Bayes Optimal Strategies for Membership Inference. In Proceed-
ings of the 36th International Conference on Machine Learning (ICML), 5558–5567.
58. A. Sablayrolles, M. Douze, Y. Ollivier, C. Schmid, and H. J´egou, “White-box vs
black-box: Bayes optimal strategies for membership inference,” 2019.
59. S. Truex, L. Liu, M. E. Gursoy, L. Yu, and W. Wei, “Towards demystifying mem-
bership inference attacks,” ArXiv, vol. abs/1807.09173, 2018.
Le-Khac, Nhien-An Black Box Attacks on Deep Anomaly Detectors, Proceedings
of the 14th International Conference on Availability, Reliability and Security
61. J. Yoon, J. Jordon, and M. van der Schaar. PATE-GAN: Generating synthetic
data with diﬀerential privacy guarantees. In International Conference on Learning
Representations, 2019. URL https://openreview.net/forum?id=S1zk9iRqF7.
62. Torkzadehmahani, R., Kairouz, P., and Paten, B. DP-CGAN: diﬀerentially private
synthetic data and label generation. CoRR, abs/2001.09700, 2020.
63. D. Arpit, S. Jastrzundeﬁnedbski, N. Ballas, D. Krueger, E. Bengio, M. S. Kanwal,
T. Maharaj, A. Fischer, A. Courville, Y. Bengio, and et al., “A closer look at
memorization in deep networks,” in Proceedings of the 34th International Conference
on Machine Learning - Volume 70, ICML’17, p. 233–242, JMLR.org, 2017.
64. S. Yeom, I. Giacomelli, M. Fredrikson, and S. Jha, “Privacy risk in machine learn-
ing: Analyzing the connection to overﬁtting,” in 2018 IEEE 31st Computer Security
Foundations Symposium (CSF), pp. 268–282, July 2018.
65. Salimans, T., Goodfellow, I.J., Zaremba, W., Cheung, V., Radford, A., Chen, X.:
Improved techniques for training gans. CoRR abs/1606.03498 (2016)
66. Xu, L., Skoularidou, M., Cuesta-Infante, A., Veeramachaneni, K.: Modeling tabular
data using conditional gan. In: NeurIPS (2019)
67. Eduardo, S., Naz´abal, A., Williams, C.K.I., Sutton, C.: Robust Variational Au-
toencoders for Outlier Detection and Repair of Mixed-Type Data. In: Proceedings
of AISTATS (2020)
68. Camino, R., Hammerschmidt, C., State, R., 2018. Generating multi-categorical
samples with generative adversarial networks. arXiv preprint arXiv:1807.01202.
Towards Improving Privacy of Synthetic DataSets 15
69. C. R. Meehan, K. Chaudhuri, and S. Dasgupta, “A non-parametric test to detect
data-copying in generative models,” ArXiv, vol. abs/2004.05675, 2020.
70. Z. Izzo, M. A. Smart, K. Chaudhuri, and J. Zou, “Approximate data deletion from
machine learning models: Algorithms and evaluations,” ArXiv, vol. abs/2002.10077,
2020.
71. Congzheng Song and Vitaly Shmatikov. Overlearning Reveals Sensitive Attributes.
In International Conference on Learning Representations (ICLR), 2020.
72. Luca Melis, Congzheng Song, Emiliano De Cristofaro, and Vitaly Shmatikov. Ex-
ploiting Unintended Feature Leakage in Collaborative Learning. In IEEE Sympo-
sium on Security and Privacy (S&P), pages 497–512. IEEE, 2019.
73. Reza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov. Member-
ship Inference Attacks Against Machine Learning Models. In IEEE Symposium on
Security and Privacy (S&P), pages 3–18. IEEE, 2017.
74. Min Chen, Zhikun Zhang, Tianhao Wang, Michael Backes, Mathias Humbert,
and Yang Zhang. When Machine Unlearning Jeopardizes Privacy. CoRR
abs/2005.02205, 2020.
75. Zheng Li and Yang Zhang. Label-Leaks: Membership Inference Attack with Label.
CoRR abs/2007.15528, 2020.
76. Klas Leino and Matt Fredrikson. Stolen Memories: Leveraging Model Memoriza-
tion for Calibrated White-Box Membership Inference. In USENIX Security Sympo-
sium (USENIX Security), pages 1605–1622. USENIX, 2020.
77. Dingfan Chen, Ning Yu, Yang Zhang, and Mario Fritz. GAN-Leaks: A Taxonomy of
Membership Inference Attacks against Generative Models. In ACM SIGSAC Con-
ference on Computer and Communications Security (CCS), pages 343–362. ACM,
2020.
78. Ahmed Salem, Yang Zhang, Mathias Humbert, Pascal Berrang, Mario Fritz, and
Michael Backes. ML-Leaks: Model and Data Independent Membership Inference
Attacks and Defenses on Machine Learning Models. In Network and Distributed
System Security Symposium (NDSS). Internet Society, 2019.
79. Jinyuan Jia, Ahmed Salem, Michael Backes, Yang Zhang, and Neil Zhenqiang
Gong. MemGuard: Defending against Black-Box Membership Inference Attacks via
Adversarial Examples. In ACM SIGSAC Conference on Computer and Communi-
cations Security (CCS), pages 259–274. ACM, 2019.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
We study membership inference in settings where assumptions commonly used in previous research are relaxed. First, we consider cases where only a small fraction of the candidate pool targeted by the adversary are members and develop a PPV-based metric suitable for this setting. This skewed prior setting is more realistic than the balanced prior setting typically considered. Second, we consider adversaries that select inference thresholds according to their attack goals, such as identifying as many members as possible with a given false positive tolerance. We develop a threshold selection designed for achieving particular attack goals. Since previous inference attacks fail in imbalanced prior settings, we develop new inference attacks based on the intuition that inputs corresponding to training set members will be near a local minimum in the loss function. An attack that combines this with thresholds on the per-instance loss can achieve high PPV even in settings where other attacks are ineffective.
Article
The medical and machine learning communities are relying on the promise of artificial intelligence (AI) to transform medicine through enabling more accurate decisions and personalized treatment. However, progress is slow. Legal and ethical issues around unconsented patient data and privacy is one of the limiting factors in data sharing, resulting in a significant barrier in accessing routinely collected electronic health records (EHR) by the machine learning community. We propose a novel framework for generating synthetic data that closely approximates the joint distribution of variables in an original EHR dataset, providing a readily accessible, legally and ethically appropriate solution to support more open data sharing, enabling the development of AI solutions. In order to address issues around lack of clarity in defining sufficient anonymization, we created a quantifiable, mathematical definition for "identifiability". We used a conditional generative adversarial networks (GAN) framework to generate synthetic data while minimize patient identifiability that is defined based on the probability of re-identification given the combination of all data on any individual patient. We compared models fitted to our synthetically generated data to those fitted to the real data across four independent datasets to evaluate similarity in model performance, while assessing the extent to which original observations can be identified from the synthetic data. Our model, ADS-GAN, consistently outperformed state-of-the-art methods, and demonstrated reliability in the joint distributions. We propose that this method could be used to develop datasets that can be made publicly available while considerably lowering the risk of breaching patient confidentiality.
Conference Paper


text:
# Huck Finn

Consider the role of lying in the book. Why is there so much lying? Is this an “immoral” book as so many early critics have said?


text:
In a series of applications my team is building we would like to test our import and export functionality which requires the respective upload and download to be automated.

The functionality is simple but tied to the UI; the end-user needs to be able to click on a button on the web UI and then the resultant export file is downloaded. With uploads, the idea is to have a local file relevant to the application and then use the web UI to upload said file and then test the import functionality.

Ideally I am looking for the ability to do this via Selenium IDE but as I understand it that is not possible.

Can this be done via Selenium WebDriver (via C#)?

If not, what is the best strategy to test this functionality? I'd like to test this as the end user and not simply pointing to some back-end URL to perform these tests.

1. The way to do it with Firefox browser

FirefoxProfile profile = new FirefoxProfile();

FirefoxDriver driver = new FirefoxDriver(profile);

driver.Get("http://path/of/your.file");

change MIME/TYPE to be correct MIME type of the file(s) you're uploading, for instance: application/pdf for PDF files, application/vnd.openxmlformats-officedocument.wordprocessingml.template for MS Word 2007+ .docx files, etc.

• File upload: it is quite enough to provide full path of the file you need to upload to SendKeys method

IWebElement element = FindElement(By.Locator.of.your.file.type.input);

2. The other way to simulate files upload/download is using Apache JMeter tool, it is free, open source and designed for load testing so you will not only be able to test functionality, but also assess performance.

• File Download: it is enough to add Save Responses to a file Post Processor to save response data to specified location

• File Upload: use "Send Files with the Request" field of the HTTP Request sampler as per image below:

See Performance testing: Upload and Download Scenarios with Apache JMeter for more detailed instructions on how to conduct your use case with JMeter

It is possible to upload the files using selenium IDE. It can be done as mentioned below.

|Command|Target|Value|

|type|target_location_path|location_of_the_file_in_your_system|

Example:

|type|id=ConsignmentCustomerInvoiceFilename|C:\Users\abc\Desktop\img1.jpg|

You can identify the "target_location_path" by manually uploading the file and then finding the xpath of the uploaded file in the web app.

Please refer below screenshot to get a better understanding.


text:
# Compute $\lim\limits_{n\to\infty}\sqrt[n]{\log\left|1+\left(\frac{1}{n\cdot\log n}\right)^k\right|}$.

Compute $$\lim\limits_{n\to\infty}\left(\sqrt[n]{\log\left|1+\left(\dfrac{1}{n\cdot\log\left(n\right)}\right)^k\right|}\right).$$ What I have: $$\forall\ x\geq 0\ :\ x- \frac{x^2}{2}\leq \log(1+x)\leq x.$$ Apply to get that the limit equals $1$ for any real number $k$.

Is this correct? Are there any other proofs?

-
You are using both notations $\log$ and $\ln$. I assume that they are the same? –  Sammy Black Nov 6 '13 at 16:48
@AntonioVargas I am using the Squeeze Theorem. –  Ahaan S. Rungta Nov 6 '13 at 17:09
@SammyBlack Yes, thanks! Edited. –  Ahaan S. Rungta Nov 6 '13 at 17:09

By Stolz-Cesaro if $(x_n)$ is a positive sequence and $$\lim_n \dfrac{x_{n+1}}{x_n} = l$$ then $$\lim_n \sqrt[n]{x_n} = l.$$
Taking as $(x_n)$ the sequence you defined, an easy calculation shows that $$\dfrac{x_{n+1}}{x_n} \rightarrow 1,$$ therefore the thesis.


text:
0 like 0 dislike
45 views
Given $T_{n} = 3^{n+1}$ . Which term is the first to exceed 20 000?
| 45 views

0 like 0 dislike
20 000= 3^(n+1)

log20 000= (n+1)log3

n= log20 000/log3 -1

= 8.01...

since 20 000 is ideally at position 8.01.., it means term 8 is slightly less than 20 000, hence term 9 will be the first, since n should be whole number.
by Diamond (82.9k points)


text:
MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 Corrected terminology: "centered hexagonal number" instead of "hexagonal number"

Is there a hexagonal, triangular, square (apart from 0 and 1)?
In other words, is there a positive integer that is simultaneously

(1) a perfect square, $n^2$, $n \ge 2$,

(2) a triangular number, $\frac{m(m+1)}{2}$, $m$ an integer,

and (3) a (centered) hexagonal number, $(p+1)^3 - p^3$, $p$ an integer?

1

# Hexagonal Triangular Squares

Is there a hexagonal, triangular, square (apart from 0 and 1)?
In other words, is there a positive integer that is simultaneously

(1) a perfect square, $n^2$, $n \ge 2$,

(2) a triangular number, $\frac{m(m+1)}{2}$, $m$ an integer,

and (3) a hexagonal number, $(p+1)^3 - p^3$, $p$ an integer?


text:
## Monday, January 9, 2012

### JCCDQBHWHCN029 Complex Numbers: Exponential Polar Form

 [original source unknown]

Introduction

Just as in my previous article, we are unable to trace the original source of this question.  This question contains an interesting link between logarithms and complex arguments.  For those interested, this link will be explored deeper at university level in the topic of Complex Analysis.

Prelude Stage 1:  Understanding the Problem

What is given?

We are given  z  which is some quotient expression with complex numbers in rectangular form.  The denominator is the square of some complex number.  The numerator is actually  -1 + i  in a “reversed” form (be careful!).

What are you supposed to do / find?
Find the modulus and prove that the argument is  5pi /12 .
Note that exact answers (using fractions, surds, logarithms, … etc but not decimals) are required, as with most maths questions at JC level.  You cannot use your calculator to show the steps of this problem, although you could numerically check your answers if you want.

What topic / sub-topic is this question on?
Modulus and argument of complex numbers, cyclic roots (nth roots of a complex number), exponential (polar) form.

Prelude Stage 2:  Planning the Method of Attack

How shall we do this?  What options are available?  Is there a short-cut?
We could expand and simplify the bottom part, and then rationalise the denominator using the technique of complex conjugates.  After further simplification, we get something in rectangular form (something like ??? + ??? i).  Then find the modulus and the argument.  This method is do-able, but it is pretty tedious.
We could also convert everything first to polar form, which is more suited for multiplication, division and powers.  There are two flavours of polar forms:
(i)   trigonometric  r (cos
q + i sin q) and
(ii)  exponential  reiq
I prefer to use the exponential polar form because it is more compact and yet it contains the same information  (r and q).  Then we can use the formulas for manipulating exponential polar forms.  This seems to be the ideal method, because we can “kill two birds with one stone”: we can handle the modulus as well as the argument calculations at one go.

Prelude Stage 3:  Execution

 figure 2 - converting to polar form

One should try to sketch or mentally visualize the complex numbers  i – 1  and  Ö3 + i  on Argand diagrams.  These complex numbers form special triangles with special angles (30º, 45º, 60º) with the horizontal axis, giving exact values for the moduli and arguments.

i – 1  forms a right-angled triangle with sides 1 and 1 and hypotenuse Ö2, making 45º with the horizontal in the left upper quadrant.  The angle turned from the real positive axis is thus 135º or 3p /4  radians

Ö3 + i  forms a right-angled triangle that makes  30º  or  p /6  radians with the positive real axis.

For checking, you can use your calculator to get the arguments, but you will get decimal answers.  After dividing by  p, you may convert the result to a fraction.  Thus you can verify the exact fraction of  p  for both arguments.

Now we proceed with the exponential polar form calculations.
 figure 3 - solving the prelude

Line #1 is the original expression which we convert to exponential form (line #2) as worked out earlier.  In line #3, we do modulus and argument (i.e. angle) calculations in a single step, with the modulus on the left of the ‘e’, and the argument on the right in brackets after the ‘i’.  The subtraction ‘–’ in the argument is because of the fraction/division in previous line.  Anything that appears below the fraction line will have their arguments negated.  The  p/6  is multiplied by 2 because of the squaring in the denominator.  All these are in accordance to these powerful exponential form rules:
·      r_1 e^(i theta_1) \cdot r_2 e^(i theta_2) = (r_1 r_2) e^(i (theta_1+ theta_2))
·      r_1 e^(i theta_1) divide r_2 e^(i theta_2) = (r_1 r_2) e^(i (theta_1- theta_2))
·      (r e^(i theta) )^n = (r^n) e^(i (n theta))

After simplifying, we obtain line #4.  We should check that  (5pi)/(12)  is in the range

(-p, p] the principal range for complex arguments.  If not, we would need to adjust by adding or subtracting some multiple of  2p.  Finally, we just pick out the modulus (line #5) and the argument (line #6) to answer the question properly.

Prelude Stage 4:  Evaluation

Are we done for this part?  Is there another way to do this?
Yes, we have answered the question as required, and we did it by combining the calculation with the proof.  We would also have done these separately, but it would have been longer to write out the solution.  This solution is short and sweet.

We can use the graphing calculator to verify that the modulus is correct.  Calculate and store the expression as ‘Z’.  Then we subtract Ö2/4  from the modulus (or absolute value) of ‘Z’ to see if we get zero.  This would show that those two expressions are equal.  The picture below shows how you can do it using Texas Instruments TI-84.  You can do something similar using the Casio fx-9860G.

 figure 4 – numerically verifying the answer using a calculator

Yes!  Got it!!  As for the proof of the argument  5p/12, we just need to check that our steps were logical.

Part (i)  -- Piece of Cake

This cyclic roots part of the question is solved by the standard method.  It is fairly straightforward, although the numbers are a bit ugly.  I shall just present my solution here below.

 figure 5 – solution to part (i)

Part (ii) Stage 1:  Understanding the Problem

What are we supposed to find?

We are to find the exact values of  a  and  b,  hence almost certainly, no decimal answers are accepted.  [This is because exact answers almost always entail an infinite number of decimal places, which our calculators are not able to supply.]  Although the  z3  term looks like we are need to solve for  z (as would be the usual case), let us remember that the  z  is actually given and known.  That is why we need to read the question carefully to avoid getting ensnared.

Part (ii) Stage 2:  Planning the Method of Attack

The question looks a bit unusual, but do not be frightened.  We observed that the RHS involves an exponential expression, which suggests the use of the exponential polar form.  As for the LHS, since  z  is known,  we can actually calculate  z3.  In what form?  Rectangular or Polar?  In exponential polar form, as suggested by the RHS and also because this form is compact and especially convenient for the calculation of powers.  Let’s try this approach and see.

Part (ii) Stage 3:  Execution

 figure 6 – solution to part (ii)

For the LHS, we use the information taken from the earlier prelude to re-express  z  in polar form (line #2).  The key observation is that on the RHS, the + in the exponential index changes to a multiplication as we split  ea+ib  into two parts.  Treating  ea  as one entity (learn to observe and recognise ‘chunks’ as one), we see that the RHS is actually in exponential polar form  reiq,  with  r = ea  and  q = b.  In line #3, we apply the exponential law  (reiq)n = r n ei (nq).  However, we note that   (5pi)/(4)   is outside the required range.  It is more than  p.  So we need to adjust it by subtracting 2p.  We need to add or subtract as many times  2p  as necessary to keep the argument between  -p  and  p.  Here, we subtract just  2p.  [ Remark: although   (5pi)/(4)  is  not the same as  -(3pi)/(4) ,   e^(i(5pi)/(4))  is  exactly the same as  e^(-i(3pi)/(4)) , because they refer to the same complex number i.e. they are represented by the same point on the Argand diagram.  The difference is how much you turn clockwise or anti-clockwise from the positive real axis to get to that point.  ]

After doing this adjustment (line #4), we are all set up to compare the moduli (highlighted in green) and the arguments (highlighted in pink) on both sides (lines #5 to #7).  We are using the fact that if   r_1 e^(i theta_1) = r_2 e^(i theta_2)   then   r_1 = r_2   and     theta_1 = theta_2 ,  provided both  q 1  and  q 2  lie in the range  (-p, p].  Remember that in maths, it is useful to be able to recognise chunks.  In this context, the relevant chunks are those that give the moduli and the arguments, as highlighted.  To uncover  a,  we take logarithms (line #6).  Line #8 just writes the answer nicely.

Part (ii) Stage 4:  Evaluation

Are we done?  Is the answer correct?
Yes, we have found the values of  a  and  b  and  b  is in the correct range,  as required.  We can verify the answers numerically using the calculator.  For example, like this:-
 figure 7 – checking part (ii) answers
Take full advantage of the graphing calculator’s ability to store complex numbers into variables like ‘Z’, ‘A’ and ‘B’.

Part (ii) Stage 5:  Reflection

What did we learn by solving this problem?
We learnt that the exponential (polar) form of a complex number is a very powerful and compact way to solve complex number problems.  It is able to handle both the modulus (distance from 0) and the argument (angle with the positive real axis) simultaneously.  It is convenient for manipulating products, quotients and powers.  We learned to apply the following rules for manipulating the exponential polar form of complex numbers.

·      r_1 e^(i theta_1) \cdot r_2 e^(i theta_2) = (r_1 r_2) e^(i (theta_1+ theta_2))
·      r_1 e^(i theta_1) divide r_2 e^(i theta_2) = (r_1 r_2) e^(i (theta_1- theta_2))
·      (r e^(i theta) )^n = (r^n) e^(i (n theta))
·     if   r_1 e^(i theta_1) = r_2 e^(i theta_2)   then   r_1 = r_2   and     theta_1 = theta_2 ,
provided both
q 1  and  q 2  lie in the range  (-p, p]

Using the last formula, we learned to compare moduli and arguments to solve for unknowns.
[ Remark: Because of Euler’s Formula eiq = cos q + i sin q, the idea behind De Moivre’s Theorem  (cos q + i sin q)n = cos (nq) + i sin (nq)  is actually subsumed by the third formula which implies that  (eiq)n = ei( nq)  when  r = 1.]

We applied metacognition (self-monitoring and self-checking) to all the 5 stages of the maths problem-solving process.  We used heuristics like making comparisons, making observations, recognising relevant chunks, comparing.  We also practised being careful by checking all our answers and verifying that we answered the question in the form required, and verifying that the complex-argument answers are within the correct range.

If you encountered a similar problem in future, would you be able to solve it?
[Please say “Yes!”, but make sure you can do it.]

Can you explain to a friend how to solve this sort of maths problem?

Can you set a similar question for yourself or your friend to solve?

#### 1 comment:

1. Yes, complex numbers are very useful, especially in exponential polar form. They are used in quantum probability computations, digital astronomical communications, and the proof of the continuum hypothesis.


text:
# A riddle about one of my favorite things :)

My source is borne upon the wing of valkyries of gold,

They work their magic summer long then winter through the cold,

Once the black and golds are done I'm plucked with careful hand,

And left to swim among the shrooms until it's time to stand,

And in that time my power grows as was the hand's intent,

Til even mighty fall like fools when my power is spent,

Though fewer know me by the days, some think I'm in the past,

My history is far from gone, it stretches wide and vast.

What am I?

Is it

"My source is borne upon the wing of valkyries of gold,

They work their magic summer long then winter through the cold,"

Bees make honey

"Once the black and golds are done I'm plucked with careful hand,"

Honey collected by beekeepers

"And left to swim among the shrooms until it's time to stand,

And in that time my power grows as was the hand's intent,"

The honey is left to ferment into mead

"Til even mighty fall like fools when my power is spent,"

"Though fewer know me by the days, some think I'm in the past,

My history is far from gone, it stretches wide and vast."

Mead was more popular historically than in the modern age

• You've pretty much got it with your third spoiler. Want to adjust the answer and I'll mark correct and add a full explanation? – Brent Hackers Jun 6 '16 at 9:43
• Sure! Gimme a sec – shree.pat18 Jun 6 '16 at 9:44
• That's pretty spot on actually. No need for a full explanation from me. +1 – Brent Hackers Jun 6 '16 at 10:05

Bit out on a limb, but:

Motorcycle

because:

My source is borne upon the wing of valkyries of gold,

They work their magic summer long then winter through the cold,

TT races (http://www.iomtt.com/), Ice Speedway

Once the black and golds are done I'm plucked with careful hand,

And left to swim among the shrooms until it's time to stand,

motorcycles sound like shroom while moving

And in that time my power grows as was the hand's intent,

throttle is controlled by hand

Til even mighty fall like fools when my power is spent,

no fuel left?

Though fewer know me by the days, some think I'm in the past,

classic bikes

My history is far from gone, it stretches wide and vast.

harley davidson

• Nope. I like those too though. – Brent Hackers Jun 4 '16 at 14:34

I am fairly certain this is about:

Beer

Once the black and golds are done I'm plucked with careful hand

refers to barley and rye

And in that time my power grows as was the hand's intent

refers to fermentation

Til even mighty fall like fools when my power is spent

is pretty obvious at this point

and might I add, Nice Riddle!

• It's not beer! Can't stand the stuff to be honest. Good explanations though they are, you haven't solved the full puzzle. (also, thanks!) – Brent Hackers Jun 4 '16 at 21:05
• Ahhh, then whiskey? It must be some sort of alcohol. I don't drink so I am not familiar with the different types, but I will keep looking. – Bryan Goggin Jun 4 '16 at 21:07
• It's not whiskey either. Not a fan of it at all. :/ – Brent Hackers Jun 4 '16 at 21:09
• Try and work out the rest of the lines. The black and golds have nothing to do with 'barley and rye'. +1 anyway though – Brent Hackers Jun 4 '16 at 21:13


text:
# A rifle of 20 kg mass can fire 4 bullets/s. The mass of each bullet is 35 × 10-3 kg and its final velocity is 400 m/s…

Q: A rifle of 20 kg mass can fire 4 bullets/s. The mass of each bullet is 35 × 10-3 kg and its final velocity is 400 m/s. Then, what force must be applied on the rifle so that it does not move backwards while firing the bullets ?

Sol: Law of conservation of momentum $\large \vec{M V} + \vec{4m v} = 0$

$\large \vec{V} = -\frac{\vec{4m v}}{M}$

$\large = -\frac{4 \times 35 \times 10^{-3}\times 400}{20}$

= -2.8 m/s

Force applied on the rifle

$\large F = \frac{MV}{t} = -\frac{20 \times 2.8}{1}$

= -56 N