# Thread: Help for representing sets

1. ## Help for representing sets

Hello,

First of all, this is my first topic in the forum. I'm from Spain and thus, my english is not very fluent. A apologize for it.

I'm trying to describe an experiment and I have some problems where describing it.

I have a set M of samples. Each sample x (in) M, is composed by the next elements:

x = { t, m, h_a, h_b, j_a, j_b, k_a, k_b}

(h_a can be viewed as h(subindex)a)

t is a value between 0 and 100
m is a value of {1,2,3,4,5}
h_a, h_b is a pair of values regarding an specific feature H
j_a, j_b is another pair regarding another specific feature J
k_a, k_b is another pair regarding another feature K

I don't know which is the best way to represent the properties of these lasts values, because for each pair of values (h, j and k) they fit these properties:

h_a < h_b and h_a + epsilon >= h_b
j_a < j_b and j_a + epsilon >= j_b
k_a < k_b and k_a + epsilon >= k_b

I mean, the a value is fewer than b, and moreover the difference between them is not greater than epsilon (epsilon is a fixed value).

I don't know how I should define each sample x. Can I define it as explained above?? There are another better way to represent it??

I also thought in representing x such as:

x (in) M, x={t,m,H,J,K} and For all C, (where C is the set H, J or K) fits that C={C_a,C_b} (C has two values) , C_a < C_b and C_a + epsilon >= C_b

I mean, representing 3 sets (H,J,K) and each one composed by two values (a and b) that fit these properties.

I hope someone could help me.

Best.

2. Unless you are seeking a representation that you can feed into a computer that understands only very limited mathematics (e.g., it does not understand quantifiers), picking a particular representation is a matter of style. It may be a little ugly, but it's a definition, and in mathematics it is not customary to argue about definitions.

I think that both of your versions are fine. I would probably use the second one because it gives more structure, i.e., it shows that h_a, h_b, j_a, j_b, k_a, k_b are not unrelated numbers but the ends of three intervals. One only needs to consider an interval as an ordered pair, not as a set. Let's denote any ordered sequence of elements $\displaystyle x_1,\dots x_n$ by $\displaystyle \langle x_1,\dots,x_n\rangle$. Then a sample is a quintuple $\displaystyle \langle t,m,h,j,k\rangle$ where $\displaystyle 0\le t\le 100$ (is $\displaystyle t$ an integer or a real?), $\displaystyle m\in\{1,2,3,4,5\}$, and each of $\displaystyle h$, $\displaystyle j$, and $\displaystyle k$ is a pair of some real numbers $\displaystyle \langle x,y\rangle$ such that $\displaystyle x\le y\le x+\epsilon$.

3. Thank you emakarov by your concrete answer.

Therefore, according your definition, I could say something like:

Given two samples $\displaystyle x_1$ and $\displaystyle x_2$ where $\displaystyle x_1=\langle t_1,m_1,h_1,j_1,k_1\rangle$ and $\displaystyle x_2=\langle t_2,m_2,h_2,j_2,k_2\rangle$, and $\displaystyle t_1<t_2$ we define a composed sample $\displaystyle c$ as a 6-tuple:

$\displaystyle c=\langle t_1,m_1,h_1,j_1,k_1,t_c,k_2\rangle$ where $\displaystyle t_c=t_2-t_1$

I think this representation should be fine.

Or would be better if I say??:

$\displaystyle c=x_1 \cup \langle t_c,k_2\rangle$ where $\displaystyle t_c=t_2-t_1$

Cheers.

4. we define a composed sample as a 6-tuple:

where
First, this is a 7-tuple. Also, since a sample was defined as a 5-tuple, a composed sample is not a sample. It is probably best to avoid such terminology because it may lead to confusion.

where
The operation \cup is defined on sets, not on ordered sequences (at least, that's not a universally accepted notation). You may introduce notation for concatenation, e.g., $\displaystyle \langle x_1,\dots x_n\rangle*\langle x_{n+1},\dots x_{n+m}\rangle=\langle x_1,\dots,x_{n+m}\rangle$; then $\displaystyle c=x_1*\langle t_c,k_2\rangle$.

Two general remarks. Different types of collections can be classified by whether (1) the order of elements and (2) the number of times an elements occurs, which is called the multiplicity of the element, matter. One must distinguish between:

(1) ordered sequences (also called tuples or lists), where both order and multiplicity matter,

(2) multisets (or bags), where the order does not matter but multiplicity does, and

(3) sets, where neither order nor multiplicity matters.

One must also make sure that expressions are well-typed. For example, if $\displaystyle \star$ is a binary operation that takes a sequence and a set, then it would be wrong to apply it to a multiset and a number, unless come convention is explicitly made.

5. Thank you so much emakarov for your remarks!

I hesitated whether to define the samples as sets or as you said, tuples. Thus, I think I was confusing some terminologies.

It's important for me to use a correct notation. My knowledge of discrete mathematics is not broad, but I think I've understood the main issues.

Thank you again

Cheers

6. Hello again,

I want to say something like:

We have a collection $\displaystyle M$ of samples $\displaystyle \in M$. A sample is a quintuple where ( is integer), , and each of , , and is a pair of some real numbers such that .

Given two different samples and where and , and $\displaystyle t_1 < t_2$.

For a pair of values $\displaystyle \langle a,b\rangle$ being $\displaystyle \langle a,b\rangle \in\{\langle h_1,h_2\rangle ,\langle j_1,j_2\rangle ,\langle k_1,k_2\rangle\}$, and $\displaystyle a=\langle a_c,a_w\rangle$, $\displaystyle b=\langle b_c,b_w\rangle$. We define the difference between $\displaystyle a$ and $\displaystyle b$ as:

$\displaystyle d(a,b)=\{\langle x,y \rangle \rightarrow x=|b_c-a_c|, y=|b_w-a_w|\}$

Given $\displaystyle x_1$ and $\displaystyle x_2$ we define $\displaystyle c_1$ and $\displaystyle z_1$ as:

$\displaystyle c_1=x_1*\langle t_c,k_2 \rangle$

$\displaystyle z_1=\langle t_1,m_1,d(h_1,h_2),d(j_1,j_2),d(k_1,k_2),t_c \rangle$

where $\displaystyle t_c=t_2-t_1$

Therefore, for each sample of $\displaystyle M$, we generate two collections $\displaystyle C=\langle c_1, c_2, \ldots c_{n-1}\rangle$ and $\displaystyle Z=\langle z_1, z_2, \ldots c_{n-1}\rangle$ for a fixed $\displaystyle t_c=1$.

Is there more or less understandable?? and the most important: is this well-formulated??

Thank you.

7. We have a collection of samples . A sample is a quintuple where ( is integer), , and each of , , and is a pair of some real numbers such that .
Omit $\displaystyle \langle x_1,\dots,x_n\rangle\in M$. You explain what a sample is later.

Given two different samples and where and , and .
It's a remark more about grammar, but this cannot be an independent sentence. You can say "Consider two samples... and $\displaystyle t_1<t_2$."
For a pair of values being , and , .
This also cannot be an independent sentence.

Now let's consider expression types (I mean pairs, tuples, sets, functions, etc.). If , then, for example, $\displaystyle \langle a,b\rangle$ can be $\displaystyle \langle h_1,h_2\rangle$. From the earlier posts it seems that $\displaystyle h_1,h_2$ are real numbers. If $\displaystyle \langle a,b\rangle=\langle h_1,h_2\rangle$, then $\displaystyle a=h_1$ and $\displaystyle b=h_2$. (Remark: this is true for tuples; if $\displaystyle \{x,y\}=\{x',y'\}$ as sets, then $\displaystyle x$ can be either $\displaystyle x'$ or $\displaystyle y'$.) Therefore, $\displaystyle a$ and $\displaystyle b$ are real numbers. However, you immediately write , so $\displaystyle a$ is a pair of something, not a number.

We define the difference between and as:

Can you define this operations on all pairs of reals, or is it important that ? If it is not important, you save a lot of space.

The right-hand side of the definition is in curly braces, so it is a set. Do you mean that $\displaystyle d(a,b)$ is a set (of something) or a pair of reals?

I would write, "We define an operation $\displaystyle d$ on pairs of real numbers as follows: $\displaystyle d(\langle x_1,y_1\rangle,\langle x_2,y_2\rangle)=\langle |x_1-x_2|,|y_1-y_2|\rangle$."

Given and
It's better to move the earlier "Given two different samples and " here. I almost forgot what and were.

Do $\displaystyle c$ and $\displaystyle z$ need ubscripts? It seems more like you define two operations in addition to $\displaystyle d$: $\displaystyle c(x_1,x_2)=x_1*\langle t_c,k_2\rangle$ and similarly for $\displaystyle z$. (Don't forget to say what $\displaystyle *$ is.)

Therefore, for each sample of , we generate two collections and for a fixed .
Above, it looked like $\displaystyle C$ and $\displaystyle Z$ (or $\displaystyle c$ and $\displaystyle z$?) were generated from two samples $\displaystyle x_1$ and $\displaystyle x_2$.

Replace "sample of " either with "sample in $\displaystyle M$" or "element of $\displaystyle M$".

The word "collection" is a synonym for "set".

Why do you write $\displaystyle c_{n-1}$ for some undetermined $\displaystyle n$? Isn't it known how many elements are in $\displaystyle C$ and $\displaystyle Z$?

I don't understand "for a fixed ". What if for some samples $\displaystyle x_1$ and $\displaystyle x_2$, $\displaystyle t_c$ is not 1?

8. Thank you emakarov for your early answer.

Originally Posted by emakarov
Now let's consider expression types (I mean pairs, tuples, sets, functions, etc.). If , then, for example, $\displaystyle \langle a,b\rangle$ can be $\displaystyle \langle h_1,h_2\rangle$. From the earlier posts it seems that $\displaystyle h_1,h_2$ are real numbers. If $\displaystyle \langle a,b\rangle=\langle h_1,h_2\rangle$, then $\displaystyle a=h_1$ and $\displaystyle b=h_2$. (Remark: this is true for tuples; if $\displaystyle \{x,y\}=\{x',y'\}$ as sets, then $\displaystyle x$ can be either $\displaystyle x'$ or $\displaystyle y'$.) Therefore, $\displaystyle a$ and $\displaystyle b$ are real numbers. However, you immediately write , so $\displaystyle a$ is a pair of something, not a number.
(1)First I define each of $\displaystyle h$, $\displaystyle j$, and $\displaystyle k$ as a pair of some real numbers $\displaystyle \langle x,y\rangle$, such that $\displaystyle x \le y \le x+\epsilon$.

Then, I say that for a pair of values $\displaystyle \langle a,b \rangle$ being $\displaystyle \langle a,b\rangle \in \{\langle h_1,h_2\rangle ,\langle j_1,j_2\rangle ,\langle k_1,k_2\rangle \}$. Therefore, if $\displaystyle \langle a,b\rangle$ are for example $\displaystyle \langle h_1,h_2\rangle$ actually $\displaystyle a$ is a pair of real numbers, and $\displaystyle b$ is another pair of real numbers. Thus, I then define $\displaystyle a$ as $\displaystyle \langle a_c,a_w\rangle$ and $\displaystyle b$ as $\displaystyle \langle b_c,b_w\rangle$, because of the first definition (1).

I think is quite confusing... how could I write it??

Originally Posted by emakarov
Can you define this operations on all pairs of reals, or is it important that ? If it is not important, you save a lot of space.

The right-hand side of the definition is in curly braces, so it is a set. Do you mean that $\displaystyle d(a,b)$ is a set (of something) or a pair of reals?

I would write, "We define an operation $\displaystyle d$ on pairs of real numbers as follows: $\displaystyle d(\langle x_1,y_1\rangle,\langle x_2,y_2\rangle)=\langle |x_1-x_2|,|y_1-y_2|\rangle$."
Yes, I want to say that for any pair of $\displaystyle \langle h_1,h_2\rangle$ or $\displaystyle \langle j_1,j_2\rangle$ or $\displaystyle \langle k_1,k_2\rangle$ (which actually are 4 real numbers) the operation "difference" can be defined as two real numbers, the first one regarding the difference of the $\displaystyle _c$ terms, and the second one regarding the difference of the $\displaystyle _w$ terms. Actually as you propose above is good, but i would to make the references using terminology $\displaystyle _c$ and $\displaystyle _w$, if possible.

Originally Posted by emakarov
Do $\displaystyle c$ and $\displaystyle z$ need ubscripts? It seems more like you define two operations in addition to $\displaystyle d$: $\displaystyle c(x_1,x_2)=x_1*\langle t_c,k_2\rangle$ and similarly for $\displaystyle z$. (Don't forget to say what $\displaystyle *$ is.)

Why do you write $\displaystyle c_{n-1}$ for some undetermined $\displaystyle n$? Isn't it known how many elements are in $\displaystyle C$ and $\displaystyle Z$?

I don't understand "for a fixed ". What if for some samples $\displaystyle x_1$ and $\displaystyle x_2$, $\displaystyle t_c$ is not 1?
Let us imagine these samples:
$\displaystyle x_1=\langle 0,1,\langle 1.8,1.83\rangle ,\langle 4, 4.5\rangle, \langle 2.11, 2.14\rangle\rangle$
$\displaystyle x_2=\langle 1,3,\langle 1.9,1.92\rangle ,\langle 4.2, 4.6\rangle, \langle 2.04, 2.08\rangle\rangle$
$\displaystyle x_3=\langle 2,3,\langle 1.89,1.91\rangle ,\langle 4.1, 4.3\rangle, \langle 2.02, 2.03\rangle\rangle$

The first value of the tuples ($\displaystyle t$) is regarding time. Therefore, supose each sample $\displaystyle x_i$ has a value of $\displaystyle i-1$ for $\displaystyle t$. Therefore, we could have 100 samples ordered by $\displaystyle t$ value.

The last three pairs are the values of $\displaystyle h$, $\displaystyle j$ and $\displaystyle k$, respectivelly.

Thus, for the first two samples $\displaystyle x_1$ and $\displaystyle x_2$, we can create:

$\displaystyle z_1=\langle t_1,m_1,d(h_1,h_2),d(j_1,j_2),d(k_1,k_2),t_c\rangl e$
as
$\displaystyle z_1=\langle 0,1,\langle 0.1,0.09\rangle ,\langle 0.2,0.1\rangle, \langle 0.07,0.06\rangle,1\rangle$

This $\displaystyle z_1$ is created using samples $\displaystyle x_1$ and $\displaystyle x_2$. Therefore, if $\displaystyle t$ changes from $\displaystyle x_i$ to $\displaystyle x_{i+1}$ in 1 unit, we will create a set of $\displaystyle n-1$ z's, being $\displaystyle n$ the number of samples of $\displaystyle M$

This is for the case that $\displaystyle t(x_1)=0,t(x_2)=1,t(x_3)=2\ldots$.

Cheers

9. Originally Posted by klendo
Then, I say that for a pair of values $\displaystyle \langle a,b \rangle$ being $\displaystyle \langle a,b\rangle \in \{\langle h_1,h_2\rangle ,\langle j_1,j_2\rangle ,\langle k_1,k_2\rangle \}$.
If you write $\displaystyle \langle a,b\rangle\in M,$ it means that $\displaystyle \langle a,b\rangle$ is an element of $\displaystyle M$, not that $\displaystyle a$ and $\displaystyle b$ separately are elements of $\displaystyle M$. Sometimes people write $\displaystyle a,b\in A$ as a contraction for "$\displaystyle a\in A$ and $\displaystyle b\in A$", but in this case they never put $\displaystyle a,b$ inside some structure like $\displaystyle \langle\cdot,\cdot\rangle$. Another way of saying that $\displaystyle a$ and $\displaystyle b$ are elements of $\displaystyle M$ is $\displaystyle \langle a,b\rangle\in M^2$, where $\displaystyle M^2=M\times M$ is the Cartesian product, but this looks a little pedantic.

I suggest introducing samples as a sequence $\displaystyle x_1,x_2,x_3,\dots$ instead of starting with a set $\displaystyle M$. You can require that the sequence is ordered by time, i.e., the first component (element) of tuples. This way readers are starting to create mental picture of a stream of data from the beginning. Besides, if each sample has a value of for , then there is no need to include time in $\displaystyle x_i$.

Then describe how to construct $\displaystyle c_i$ and $\displaystyle z_i$ from $\displaystyle x_i$ and $\displaystyle x_{i+1}$. It's better to describe this construction for an arbitrary $\displaystyle i$th element instead of $\displaystyle x_1$ and $\displaystyle x_2$.

Now, I realize that in the posts above, $\displaystyle x_1$ and $\displaystyle x_2$ were introduced as arbitrary elements of $\displaystyle M$, not the first and second elements of the sequence. But if in the end you apply different operations ($\displaystyle d$, constructing $\displaystyle c_i$ and $\displaystyle z_i$) only to samples with time difference 1, then there is no need to talk about arbitrary $\displaystyle x_1$ and $\displaystyle x_2$, and then require that they are neighbors.

Briefly, either define all constructions for two arbitrary samples and then don't expect that $\displaystyle t_2-t_1=1$, or introduce samples as a sequence of observations and talk about $\displaystyle x_i$ and $\displaystyle x_{i+1}$.

10. The question is that $\displaystyle t$ is not going to be 1, it can be 1, 2, 5 or 10. Thus, the opperation $\displaystyle d$ can involve samples which are not neighbours.

A sample is a quintuple where is some integer value such that , , and each of , , and is a pair of some real numbers $\displaystyle \langle c,w\rangle$ such that $\displaystyle c\le w\le c+\epsilon$. $\displaystyle c$ represents the c-value of the specific component, and $\displaystyle w$ represents the w-value. COMMENTS: I think that introduce c and w here avoids me to simplify the next definitions

The $\displaystyle t$ component is regarding time. Samples are taken every minute. Therefore we have a collection $\displaystyle \langle x_1,x_2,x_3,\ldots x_n\rangle$ of samples and for every pair of samples $\displaystyle x_i$ and $\displaystyle x_{i+1}$ $\displaystyle \rightarrow t(x_i)<t(x_{i+1})$. COMMENTS: Can I write this?? or it would be better to define the components of both samples??. Another issue to take into account is that $\displaystyle t(x_{i+1})$ may not be $\displaystyle t(x_i)+1$, because if we take samples for each minute (1,2,3, etc.), if the sample regarding the minute 4 do not fulfill that $\displaystyle y\le x+\epsilon$ this sample is not stored, thus the collection of samples may be ordered by time such as (1,2,3,5,6...).

If you write it means that is an element of , not that and separately are elements of . Sometimes people write as a contraction for " and ", but in this case they never put inside some structure like . Another way of saying that and are elements of is , where is the Cartesian product, but this looks a little pedantic.
COMMENT: Actually I only apply $\displaystyle d$ for the pairs $\displaystyle h_1,h_2$, $\displaystyle j_1,j_2$ and $\displaystyle k_1,k_2$, but I could define d as you suggest:

We define an operation on pairs of real numbers as follows: . COMMENTS: Actually, the x elements are regarding the c-value and the y elements are regarding the w-value, I suppose it would be better to replace x and y for c and w.

Given two different samples $\displaystyle x_i=\langle t_i,m_i,h_i,j_i,k_i\rangle$ and $\displaystyle x_{j}=\langle t_j,m_j,h_j,j_j,k_j\rangle$, such that $\displaystyle t_i < t_j$. We define $\displaystyle z_i=\langle t_i,m_i,d(h_i,h_j),d(j_i,j_j),d(k_i,k_j),t_c\rangl e$ where $\displaystyle t_c=t_j-t_i$, and also define $\displaystyle c_i=\langle t_i,m_i,h_i,j_i,k_i,t_c,k_j\rangle$.

For every sample $\displaystyle x_i$, we find a different sample $\displaystyle x_j$ such that $\displaystyle t_i+1=t_j$ and construct a $\displaystyle Z$ and $\displaystyle C$ collections composed by every $\displaystyle c_i$ and $\displaystyle z_i$. This process is repeated for every pair of samples $\displaystyle x_i$ and $\displaystyle x_j$ such that:
$\displaystyle t_i+2=t_j$
$\displaystyle t_i+5=t_j$
$\displaystyle t_i+10=t_j$

What do you think??

I'm sorry to cause you so much trouble.

Thank you.

11. Originally Posted by klendo
A sample is a quintuple where is some integer value such that , , and each of , , and is a pair of some real numbers $\displaystyle \langle c,w\rangle$ such that $\displaystyle c\le w\le c+\epsilon$. $\displaystyle c$ represents the c-value of the specific component, and $\displaystyle w$ represents the w-value.
This sounds good. If you are going to explain what $\displaystyle c$ and $\displaystyle w$ are, you can also say that $\displaystyle t$ represents time in seconds; then you can refer to it, e.g., as "the time component of $\displaystyle x_i$".

Therefore we have a sequence $\displaystyle \langle x_1,x_2,x_3,\ldots x_n\rangle$ of samples and for every pair of samples $\displaystyle x_i$ and $\displaystyle x_{i+1}$ $\displaystyle \rightarrow t(x_i)<t(x_{i+1})$. COMMENTS: Can I write this?? or it would be better to define the components of both samples??
Generally, you need to define what $\displaystyle t(x_i)$ means. There are multiple ways to rephrase this.

• Therefore we have a sequence of samples ordered by time i.e., by the first component.
• Therefore we have a sequence of samples $\displaystyle x_1,x_2,x_3,\ldots x_n$ ordered in such a way that if $\displaystyle i<j$, $\displaystyle x_i=\langle t_i,\dots\rangle$, $\displaystyle x_j=\langle t_j,\dots\rangle$, then $\displaystyle t_i<t_j$.
• Therefore we have a sequence of samples $\displaystyle x_1,x_2,x_3,\ldots x_n$. To simplify notation, if $\displaystyle x_i=\langle t,m,h,j,k\rangle$, we write $\displaystyle t_i$ to refer to $\displaystyle t$, and similarly for other components of $\displaystyle x_i$. It is assumed that the samples are ordered by time, i.e., $\displaystyle t_i<t_j$ whenever $\displaystyle i < j$.

We define an operation on pairs of real numbers as follows: . COMMENTS: Actually, the x elements are regarding the c-value and the y elements are regarding the w-value, I suppose it would be better to replace x and y for c and w.
Yes, do this.

Given two different samples and , such that . We define...
This, as well as "If ..., then ..." cannot be broken into two sentences. (I made this mistake in the past.) For two sentences, start with "Assume", "Consider", "Let", "Suppose" and the like.

For every sample $\displaystyle x_i$, we find a different sample $\displaystyle x_j$ such that $\displaystyle t_i+1=t_j$
This is an important point: You should not write a sentence that is wrong when considered by itself and that, as a consequence, require further text to impose some restrictions, give an explanation, make an addition, etc. Every statement must be true in the place where it is written.

What you say here implies that $\displaystyle t_{i+1}-t_i=1$ for all $\displaystyle i$, which is not correct according to what you said above. In such case, one should either describe the general case, give an example (and tell the readers about it), or at least write a remark that further details will be provided later.

In general, try to anticipate what the reader thinks when he/she reads your text. If readers know that samples can be separated by more than a second and reads the claim above, there is going to be some head scratching until they reach $\displaystyle t_i+2=t_j$.

For every sample $\displaystyle x_i$, we find a different sample $\displaystyle x_j$ such that $\displaystyle t_i+1=t_j$
and construct a $\displaystyle Z$ and $\displaystyle C$ collections composed by every $\displaystyle c_i$ and $\displaystyle z_i$.
One should say "collections $\displaystyle Z$ and $\displaystyle C$". A phrase "$\displaystyle Z$ collection" or "Z-collection" makes me think about special type of collections called Z-collections instead of a collection denoted by the letter $\displaystyle Z$.

Here, I think, the fact that $\displaystyle t_i+1=t_j$ has to be explicitly tied with the way you construct $\displaystyle c_i$. The picture is the following. In the previous paragraph, you defined $\displaystyle c_i$ and $\displaystyle c_j$ for every $\displaystyle i$ and $\displaystyle j$. The scope of $\displaystyle i$ and $\displaystyle j$ closed when the paragraph ended and one cannot refer to them anymore.

Then, in this paragraph you introduce some other $\displaystyle i$ and $\displaystyle j$ and then refer to $\displaystyle c_i$. But what $\displaystyle j$ is used to construct $\displaystyle c_i$? Even though you have a $\displaystyle j$ in the current paragraph, it is not clear how it related to $\displaystyle j$ from the previous paragraph.

Maybe it's a good idea to label $\displaystyle c_i$ and $\displaystyle z_i$ with a superscript that shows $\displaystyle t_j-t_i$. Something like the following.

Given two samples $\displaystyle x_i=\langle t_i,m_i,h_i,j_i,k_i\rangle$ and $\displaystyle x_{j}=\langle t_j,m_j,h_j,j_j,k_j\rangle$ such that $\displaystyle t_j-t_i=\delta$ for $\displaystyle \delta\in\{1,2,5,10\}$, we define $\displaystyle z_i^\delta=\langle t_i,m_i,d(h_i,h_j),d(j_i,j_j),d(k_i,k_j)\rangle$ and $\displaystyle c_i^\delta=\langle t_i,m_i,h_i,j_i,k_i,t_c,k_j\rangle$. Let $\displaystyle Z^\delta=\{z_i^\delta\mid z_i^\delta***\}$, and similarly for $\displaystyle C^\delta$. (*** should read "is defined". I don't know why \hbox and \mbox don't work today.)

12. Thank you so much for your help, emakarov.

I'll try to finish my work.

Thank you.

13. I'm here again. Happy new year!

I'm already finishing the experiment description and below I'll briefly show it. What do you think about the formalism correctness?? There is any error in the specification??

A sample is a quintuple where:
- represents the time in seconds. Is some integer value such that
- ...
- represents...

Each of , , and is a pair of some real numbers such that . represents the c-value of the specific component, and represents the w-value.

.....

Therefore we have a sequence of samples. To simplify notation, if , we write to refer to , and similarly for other components of . It is assumed that the samples are ordered by time, i.e., $\displaystyle t_i<t_j$ whenever $\displaystyle i<j$

.......

Given two different samples and , such that $\displaystyle m_j-m_i=\delta$ we define a case of the case base as:

.........and .

Every pair of samples $\displaystyle x_i$ and $\displaystyle x_j$ which $\displaystyle m_j-m_i=\delta$, allows us to create the case base $\displaystyle Z^\delta=\{z_i^\delta | z_i^\delta \mbox{ is d efined} \}$. ......

Step 1: Let an input problem and a case of the case base $\displaystyle z_r^\delta=\langle t_r,m_r,h_r^\delta,j_r^\delta,k_r^\delta\rangle$, where $\displaystyle j_r^\delta=\langle c_{j_r^\delta},w_{j_r^\delta}\rangle$, $\displaystyle k_r^\delta=\langle c_{k_r^\delta},w_{k_r^\delta}\rangle$......
COMMENTS: I need to access to each element of the pair which composes $\displaystyle j_r^\delta$ and $\displaystyle k_r^\delta$, so I used this notation for representing the c and w component.

........

Step 3: From each case $\displaystyle z_r^\delta$ selected from step 2 such that $\displaystyle z_r^\delta=\langle t_r,m_r,h_r^\delta,j_r^\delta,k_r^\delta\rangle$ where $\displaystyle j_r^\delta=\langle c_{j_r^\delta},w_{j_r^\delta}\rangle$, $\displaystyle k_r^\delta=\langle c_{k_r^\delta},w_{k_r^\delta}\rangle$, we calculate:

$\displaystyle e(c,j,k)=c_{k_r^\delta}-c_{j_r^\delta}$
$\displaystyle e(w,jk)=w_{k_r^\delta}-w_{j_r^\delta}$

Then, we calculate an average for each one of these two values using all the cases selected from step 2:
$\displaystyle Average(e)=\frac{1}{N}\Sigma_{i=1}^Ne_i$, being $\displaystyle N$ the number of cases selected from the step 2.

What do you think about this?? Do you see so much errors??

Thank you.

14. Happy new year!
Thanks, and the same to you!
Given two different samples and , such that
Before, we were using $\displaystyle t_i$ and $\displaystyle t_j$ here, not $\displaystyle m$'s. It should be said that $\displaystyle t_i<t_j$ (or $\displaystyle i < j$). Also, $\displaystyle \delta$ was not introduced properly: is it some $\displaystyle \delta$? for all $\displaystyle \delta$?
Every pair of samples and which , allows us to create the case base
Again, not clear which particular $\displaystyle \delta$ you are talking about. Next, your phrasing suggests that that for each pair $\displaystyle x_i, x_j$ a possibly different $\displaystyle Z^\delta$ is generated. No, a single $\displaystyle Z^\delta$ is generated from all pairs $\displaystyle x_i,x_j$ such that $\displaystyle t_j-t_i=\delta$.

Step 1: Let an input problem and a case of the case base , where , ......
No relationship between $\displaystyle i$ and $\displaystyle r$? Why not say $\displaystyle z_r^\delta=\langle t,m,\langle c',w'\rangle,\langle c'',w''\rangle,\langle c''',w'''\rangle\rangle$? Since it is said to be a $\displaystyle z_r^\delta$, we know that it is produced from $\displaystyle x_r$ and $\displaystyle x_{r+\delta}$ and one does not need to use the indices $\displaystyle r,\delta$ everywhere.

Then, we calculate an average for each one of these two values using all the cases selected from step 2:
,
What is $\displaystyle e_i$? It was not introduced.

In general, I am not sure this is heading in the right direction. First, it becomes rather complicated to follow even for an experienced reader. Second, this difficulty level apparently is more than what you are comfortable with. I mean this: if I present my work (not just some home assignment from a marginal course I don't care about), I am passionate about it. It is my creation and I know it better than anybody in the room. If someone points a possible error, I am sure I know how to determine if this is indeed a problem because everything written originates with me; I know precisely what I have written and what I wanted to communicate. If, on the other hand, I am saying something that I saw in a book but do not fully understand, I am able neither to defend it properly nor correct it if needed.

I think it is valuable when one presents something that he/she fully understands and takes responsibility for. If I study French, I would rather write a story exclusively with "subject, verb, complement" phrases that I know how to form instead of using more complicated constructions I am not sure about. In mathematics, further, boring and repetitive style is a far less serious problem that an actual error.

A good way to grasp this thing is to write a computer program that analyzes the results of your experiment. Then you will have to form sequences, pairs, and other objects in your program, and the compiler will tell you if you have an undeclared variable, for example. Once you explain this to a computer, you can assume that you understand the subject.

15. Thank you emakarov for your patience.

Originally Posted by emakarov
Before, we were using $\displaystyle t_i$ and $\displaystyle t_j$ here, not $\displaystyle m$'s. It should be said that $\displaystyle t_i<t_j$ (or $\displaystyle i < j$). Also, $\displaystyle \delta$ was not introduced properly: is it some $\displaystyle \delta$? for all $\displaystyle \delta$?
Right. Copy-paste betrayed me. It's $\displaystyle t_i$, not $\displaystyle m_i$. I didn't define $\displaystyle \delta$ because I introduced a sentence such as: "while measuring... for 1 minute ($\displaystyle \delta=1$) we need to use two samples $\displaystyle x_i$ and $\displaystyle x_j$ such that $\displaystyle t_j-t_i=1$. Maybe it would be better to define the specific values of $\displaystyle \delta$ as $\displaystyle \delta=\{1,2,5,10\}$

[QUOTE=emakarov;435066]
Again, not clear which particular $\displaystyle \delta$ you are talking about. Next, your phrasing suggests that that for each pair $\displaystyle x_i, x_j$ a possibly different $\displaystyle Z^\delta$ is generated. No, a single $\displaystyle Z^\delta$ is generated from all pairs $\displaystyle x_i,x_j$ such that $\displaystyle t_j-t_i=\delta$.[\quote]

Right

Originally Posted by emakarov
No relationship between $\displaystyle i$ and $\displaystyle r$? Why not say $\displaystyle z_r^\delta=\langle t,m,\langle c',w'\rangle,\langle c'',w''\rangle,\langle c''',w'''\rangle\rangle$? Since it is said to be a $\displaystyle z_r^\delta$, we know that it is produced from $\displaystyle x_r$ and $\displaystyle x_{r+\delta}$ and one does not need to use the indices $\displaystyle r,\delta$ everywhere.
Right, there is not relationship. We first create a case base $\displaystyle Z^\delta$. Then, a different sample which was not used for building the case base is introduced in the system. I used $\displaystyle x_i$ to define it, but it may confuse the reader. I think there's no reason to use $\displaystyle x_i$ here. I think I'll write a sample $\displaystyle x$.

Originally Posted by emakarov
What is $\displaystyle e_i$? It was not introduced.
Ok. In step 2 I can select more than one case which fulfills the properties. Then, for each case I calculate in Step 3
$\displaystyle e(c,j,k)=c_{k_r^\delta}-c_{j_r^\delta}$
$\displaystyle e(w,j,k)=w_{k_r^\delta}-w_{j_r^\delta}$

But this is calculated for each sample... actually I need an averaged value, so I used $\displaystyle Average(e)$ to refer both, $\displaystyle e(c,j,k)$ and $\displaystyle e(c,j,k)$.

Originally Posted by emakarov
In general, I am not sure this is heading in the right direction. First, it becomes rather complicated to follow even for an experienced reader. Second, this difficulty level apparently is more than what you are comfortable with. I mean this: if I present my work (not just some home assignment from a marginal course I don't care about), I am passionate about it. It is my creation and I know it better than anybody in the room. If someone points a possible error, I am sure I know how to determine if this is indeed a problem because everything written originates with me; I know precisely what I have written and what I wanted to communicate. If, on the other hand, I am saying something that I saw in a book but do not fully understand, I am able neither to defend it properly nor correct it if needed.

I think it is valuable when one presents something that he/she fully understands and takes responsibility for. If I study French, I would rather write a story exclusively with "subject, verb, complement" phrases that I know how to form instead of using more complicated constructions I am not sure about. In mathematics, further, boring and repetitive style is a far less serious problem that an actual error.

A good way to grasp this thing is to write a computer program that analyzes the results of your experiment. Then you will have to form sequences, pairs, and other objects in your program, and the compiler will tell you if you have an undeclared variable, for example. Once you explain this to a computer, you can assume that you understand the subject.
I understand your point of view. I appreciate your comments. I could have described my work without using mathematic formalism. However, it would have been a poor and difficult to understand work. I know what are the details of the work, however I want to describe it in the most math-correctly and understandable way.

Thanks again.