derivatives

CSGnet Archive CSG2004
1

[From Bill Powers (2004.09.17.0931 MDT)]

For the record, here are some comments on functions and derivatives.

The statement y = f(t) says that that value of y is some function of time.
The notation f(t) means the value of the function at time t.

The rate of change or derivative of y with respect to time, or dy/dt, is
computed as

                   f(t2) - f(t1)
dy/dt = lim ---------------
         t2 --> t1 t2 - t1

In words, this says that the derivative is the limit of the value of the
function at time 2 minus the value at time 1, divided by the time
difference, t2 - t1. The limit is computed as the time difference
approaches zero. The numerator and denominator both approach zero as t2
approaches t1, so the question is whether the quotient remains defined all
the way to zero time difference. The kinds of functions we are usually
interested in are such that the quotient has a specific value no matter how
small the time difference is. The real definition includes all sorts of
conditions and specifications that the function has to meet to make the
above definition work, the main one being that the function must be
continuous, which is a bit circular since that means it has a finite
derivativo over its whole range of values.

When we use finite numbers in the calculation above, we are only
approximating the first derivative. But normally that's close enough for
government work.

Best,

Bill P.

2

[From Rick Marken (2004.09.17.0945)]

Bill Powers (2004.09.17.0931 MDT)]

For the record, here are some comments on functions and derivatives.

Thanks, Bill. Very nice and clear.

But the derivative thing was just a red herring used to divert attention
from the data (posted again below) that shows the budget balance being
apparently controlled during the Clinton administration and then quickly
going out of control during Bush II. Control of the budget balance during
the Clinton administration is evidenced by the fact that the balance moves
smoothly and linearly into surplus, despite variations in growth rate.
Growth rate (the black trace) is computed in the standard way:
[GNP(t)/GNP(t-1)]-1. If people think the data look the way it does because I
calculated growth rate incorrectly, then they should explain (to me, the
government and the business reporting media) how to do it correctly.

Best regards

Rick

dGNPDeficit2.gif

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3

From [Marc Abrams (2004.09.17.1641)

[From Bill Powers (2004.09.17.0931 MDT)]

For the record, here are some comments on functions and derivatives.

The statement y = f(t) says that that value of y is some function of time.
The notation f(t) means the value of the function at time t.

The rate of change or derivative of y with respect to time, or dy/dt, is
computed as

                   f(t2) - f(t1)
dy/dt = lim ---------------
         t2 --> t1 t2 - t1

Sorry Bill, I guess it's been awhile, try this; f'(x) (that's f prime of x)=

         Lim f(x + h) - f(x)
f'(x) = x ->0 ---------------
                      h ;Where h is very small change in x

A derivative is _NOT_ a change in y with respect to 'time'. It's the
_INSTANTANEOUS_ rate of change, i.e. the slope, for a tangent line on a
continuous point of a curve, for any mathematical function and it's a change
in x relative to y. The x axis is often used as a time line, but not always.

This is the reason Forrester says the derivative is a useless practical
tool. _NOTHING_ in the real world happens instantaneously.

In words, this says that the derivative is the limit of the value of the
function at time 2 minus the value at time 1, divided by the time
difference, t2 - t1. The limit is computed as the time difference
approaches zero.

Again, time is not important. The derivative is simply the _INSTANTANEOUS_
rate of change between x and y.

The numerator and denominator both approach zero as t2
approaches t1, so the question is whether the quotient remains defined all
the way to zero time difference.

No Bill, it doesn't matter if the denominator approaches zero. The whole
point of the limit is that it includes all points _EXCEPT_ the integer that
would make it zero and hence undefined.

The kinds of functions we are usually
interested in are such that the quotient has a specific value no matter
how small the time difference is.

Huh? What the hell does this mean? With the limit, you are not interested in
what the quotient is. You're interested in what the _limit_ is. _NOT_ the
same thing

The real definition includes all sorts of
conditions and specifications...

Really? Besides being 'continuous' which also implies that both limits exist
and it is indeed a function, what other 'conditions' and 'specifications'
are required for a derivative? Granted, each type of function has it's own
quirks, but to say there are "all sorts of..." is kind of stretching it a
bit.

When we use finite numbers in the calculation above, we are only
approximating the first derivative. But normally that's close enough for
government work.

All numbers are 'finite', _all_ measurements are approximations. So what?

Marc

4

[From Rick Marken (2004.09.17.1400)]

Marc Abrams (2004.09.17.1641)

Bill Powers (2004.09.17.0931 MDT)]

The rate of change or derivative of y with respect to time, or dy/dt, is
computed as
                   f(t2) - f(t1)
dy/dt = lim ---------------
         t2 --> t1 t2 - t1

Sorry Bill, I guess it's been awhile, try this; f'(x) (that's f prime of x)=

       Lim f(x + h) - f(x)
f'(x) = x ->0 ---------------
                    h ;Where h is very small change in x

A derivative is _NOT_ a change in y with respect to 'time'. It's the
_INSTANTANEOUS_ rate of change, i.e. the slope, for a tangent line on a
continuous point of a curve, for any mathematical function and it's a change
in x relative to y. The x axis is often used as a time line, but not always.

If you passed your calculus course then that's another strike against "no
child left behind". You should have been left behind to try again.

Bill's definition of the derivative is correct, yours is nonsense. An
equivalent way to write Bill's version is:

                    f(y + h) - f(y)
dy/dt = lim ---------------
          h --> 0 h

In Bill's formula, t2 = y + h and t1 = y so t2 - t1 = h.

Note that in your version, you have the argument of the function (x in your
case) approaching 0, which would mean that the derivative of f(x) exists
only when x = 0. That definition of a derivative would have gotten physics
precisely nowhere.

RSM

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5

[From Bill Powers (2004.09.17.1603 MDT)]

Rick Marken (2004.09.17.1400)--

Marc the Sharc snaps:
> Sorry Bill, I guess it's been awhile, try this; f'(x) (that's f prime of x)=

>
> Lim f(x + h) - f(x)
> f'(x) = x ->0 ---------------
> h ;Where h is very small change in x
>
> A derivative is _NOT_ a change in y with respect to 'time'. It's the
> _INSTANTANEOUS_ rate of change, i.e. the slope, for a tangent line on a
> continuous point of a curve, for any mathematical function and it's a
change
> in x relative to y. The x axis is often used as a time line, but not
always.

Gosh, you don't say!

Rick replies:

Bill's definition of the derivative is correct, yours is nonsense. An
equivalent way to write Bill's version is:

                    f(y + h) - f(y)
dy/dt = lim ---------------
          h --> 0 h

In Bill's formula, t2 = y + h and t1 = y so t2 - t1 = h.

You're both fishing around trying to score points and being hasty (which
you can also spell with an "n"). I wrote specifically about derivatives
with respect to time, where t is the customary designation for time. You
can, of course, differentiate a function with respect to any variable in
it; for example, if y = x^2, then dy/dx = 2x.

The notation f' for the first derivative of a function of one variable is
handy when you aren't writing an equation. You can refer to the derivative
(or higher derivatives, f'', f''' etc.) without mentioning what the
variable is, and it's easier to write than dy/dt or dy/dx. Engineers use
dots over the variable symbol with the same meaning, but you can't type
them on an American keyboard. On the other hand, if you mean to discuss
time derivatives (that was the subject of this series of posts), the
notation dy/dt (or df/dt if you're not writing an equation) is more explicit.

I approve of getting a mathematical education. However, students, and
critics of students, should remain meek until they have a few courses and
some experience behind them.

Bill P.

6

[From Rick Marken (2004.09.17.1600)]

Bill Powers (2004.09.17.1603 MDT)]

Rick replies:

Bill's definition of the derivative is correct, yours is nonsense. An
equivalent way to write Bill's version is:

                    f(y + h) - f(y)
dy/dt = lim ---------------
          h --> 0 h

In Bill's formula, t2 = y + h and t1 = y so t2 - t1 = h.

You're both fishing around trying to score points and being hasty (which
you can also spell with an "n").

Sorry, you're right. I was being hasty. Change y to t on the right.

What points am I trying to score? Marc keeps saying that the data I present
on GNP growth rate over time is not showing GNP growth rate over time. I
guess I could just leave it at that, but that leaves a pretty large error
signal for me. I think I'm plotting estimates of dGNP/dt over time, don't
you?

Best

Rick

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Home: 310 474 0313
Cell: 310 729 1400

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7

[From Bill Powers (2004.09.17.1716 MDT)]

Rick Marken (2004.09.17.1600)--

What points am I trying to score? Marc keeps saying that the data I present
on GNP growth rate over time is not showing GNP growth rate over time.

Since when do you judge whether you are right or wrong on the basis of what
Marc says? All he has to do is say you're wrong, and you can't resist
pushing back. Why be such a pushover?

I guess I could just leave it at that, but that leaves a pretty large error
signal for me.

I guess you believe he is a pretty deep thinker and are afraid he might be
right. Is that the error?

I think I'm plotting estimates of dGNP/dt over time, don't you?

The important thing is, don't you? Have you let an overenthusiastic
beginner shake your confidence? Look at what you wrote and what you
plotted. Is it right? If you're not sure, get a beginning calculus book and
look it up. You don't need my opinion or Marc's. Opinions aren't what make
your calculations right or wrong.

Best,

Bill P.

8

From [Marc the Sharc (2004.09.17.2003)]

[From Bill Powers (2004.09.17.1603 MDT)]

Gosh, you don't say!

Actually you didn't say. You're the one who likes to be specific and
non-fuzzy. Can't take it when it's thrown back at you can you?

Did you also forget to read this stuff, or was your chest so bloated out in
indignation that were blind to it, I said;

A derivative is _NOT_ a change in y with respect to 'time'. It's the
_INSTANTANEOUS_ rate of change, i.e. the slope, for a tangent line on a
continuous point of a curve, for any mathematical function and it's a change
in x relative to y. The x axis is often used as a time line, but not always.

This is the reason Forrester says the derivative is a useless practical
tool. _NOTHING_ in the real world happens instantaneously.

I see you choose not to comment on this, nor on any other _CORRECTION_ I
made.

>Rick replies:

>Bill's definition of the derivative is correct, yours is nonsense. An
>equivalent way to write Bill's version is:
>
> f(y + h) - f(y)
>dy/dt = lim ---------------
> h --> 0 h
>
>In Bill's formula, t2 = y + h and t1 = y so t2 - t1 = h.

Mine is _NONSENSE_ Rick? You are a complete and utter ass. Your hubris knows
no bounds and you know less about mathematics than I actually thought you
did. You don't know crap. Ask your friend from the UK. Someone really needs
to straighten your ass out. You don't even know what a difference equation
looks like

You're both fishing around trying to score points and being hasty (which
you can also spell with an "n"). I wrote specifically about derivatives
with respect to time, where t is the customary designation for time. You
can, of course, differentiate a function with respect to any variable in
it; for example, if y = x^2, then dy/dx = 2x.

So how am I fishing and being hasty when you parrot what I said above?

I approve of getting a mathematical education. However, students, and
critics of students, should remain meek until they have a few courses and
some experience behind them.

You're so full of shit it smells all the way to N.Y.C.

It really would be too much for you too admit that I have a point or two and
that Marken is _DEAD_ wrong, instead you come out with this Marc the Sharc
crap and weasel your way through a _Pathetic_ excuse for why you were so
sloppy in using mathematical terms.

Go back into your cave.

Marc

9

From [Marc Abrams (2004.09.17.1403)]

[From Rick Marken (2004.09.17.0945)]

But the derivative thing was just a red herring used to divert attention
from the data (posted again below) that shows the budget balance being

Nonsense. You can show whatever trends you want to show with the GNP, all
you have to do is _ARBITRARILY_ decided which time periods to use.

Your analysis, like you mathematics is _BOGUS_.

I tried explaining to you that a rate is not necessarily a derivative, yet
you continued to call it one. Jay Forrester explains quite clearly why a
_derivative_ is not found in the real world and why integrals are really the
accumulations of rates over time.

But this is way over your head, so stick to your 'formulas.'

Marc

10

From [Marc Abrams (2004.09.17.2025)]

[From Bill Powers (2004.09.17.1716 MDT)]

Rick Marken (2004.09.17.1600)--

You two were made for each other and you both sow what you reap.

Powers you bet your sweet ass I'm a thinker, it's one of the reasons you
can't quite come to read my posts. The truth _always_ stings.

But your personal attacks on me are meaningless, and at this point because I
truly know how worthless your comments and thoughts are, are kind of
laughable.

Your remarks are the words of a bitter old fool, who still hasn't mentally
gotten past his 5th birthday hissy fit for not getting what you wanted.

Since when do you judge whether you are right or wrong on the basis of
what
Marc says? All he has to do is say you're wrong, and you can't resist
pushing back. Why be such a pushover?

Listen to this wise counsel Rick, especially when you have Powers standing
behind you with all the 'facts'. Like how he has corrected me with all my
'mistaken' ideas and comments.

> I guess I could just leave it at that, but that leaves a pretty large
error
>signal for me.

I guess you believe he is a pretty deep thinker and are afraid he might be
right. Is that the error?

> I think I'm plotting estimates of dGNP/dt over time, don't you?

Rick, I never said you were _NOT_ plotting the _AMOUNT_ of change in the GNP
over time. You _WERE_. What you were _NOT_ doing is the _DERIVATIVE_ and I
also said that the data can show whatever 'trends' you want, all you need to
do is _arbitrarily_ pick the time frame and you can show all sorts of
stuff. Just because you decided to pick this time period, does not mean that
the numbers you are seeing are telling you what you think they are. How do
these numbers compare throughout history? Why do you think the trends of
these 15 quarters are representative of our entire history? What
significance is there in the magnitude of the numbers and has history borne
out your analysis?

Yeah I know Rick, I'm not much of thinker compared to Powers, but I don't
see him asking any serious questions.

The important thing is, don't you? Have you let an overenthusiastic
beginner shake your confidence? Look at what you wrote and what you
plotted. Is it right? If you're not sure, get a beginning calculus book
and
look it up. You don't need my opinion or Marc's. Opinions aren't what make
your calculations right or wrong.

Right-o Bill, wise advice. Look at a calculus text. What a _novel_ idea. I
bet you did _AFTER_ you read my post, and you came back with your gibberish.
It might be a good idea for Rick to look at a beginners guide to economics
while he's at it. You might want to split the costs Bill and read it after
Rick does.
And when _EITHER_ of you can point out _EXACTLY_ where I am mistaken in my
comments it might be interesting. So far, _neither_ of you have said a thing
of value. You attacked me personally, which is your modus operendi, but you
haven't said anything of substance. You're turning out to be a bigger frauds
then I originally suspected. Jim Jones is looking better and better every
day.

What is truly amazing is that you show so little regard and respect for
others and expect so much back in return. I wouldn't waste my spit on your
shoes.

Marc the Sharc

11

[From Rick Marken (2004.09.17.1900)]

Bill Powers (2004.09.17.1716 MDT)]

Rick Marken (2004.09.17.1600)--

What points am I trying to score? Marc keeps saying that the data I
present
on GNP growth rate over time is not showing GNP growth rate over time.

Since when do you judge whether you are right or wrong on the basis of
what
Marc says?

Since never. But this is a discussion group, and I think it's helpful
to discuss things.

I think I'm plotting estimates of dGNP/dt over time, don't you?

The important thing is, don't you?

Of course. But if all that's important is what one thinks of their
ideas themselves, then why have a discussion group at all?

Regards

Rick

···

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marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400

12

From [Marc Abrams (2004.09.17.2305)]

[From Rick Marken (2004.09.17.1900)]

Since never. But this is a discussion group, and I think it's helpful
to discuss things.

Discuss what? How brilliant you are? Or maybe how insightful Powers is?

What a crock. Neither one of you is man enough to admit that you might have
made a mistake. Instead you come out and try to attack my credibility, when
_any_ first year calculus student would know and recognize my argument as
being the correct one.

But rather than try and come to some understanding, by using a text if
necessary, you blather;

Bill's definition of the derivative is correct, yours is nonsense. An...

What a fool you are. I took my equation right out of my _old_ calculus text
book. So I guess Powers not only knows more about behavior than any
psychologist, he also happens to know more economics than any living
economist and more of the Calculus than any mathematician.

What a true Renaissance man he is. I hope he actually gets back to the 21st
century some day.

Yet he readily admits;

"I'm a bit hazy about the details here; it's been some years since I
reviewed the facts. At any rate, it takes a certain amount of charge to
change"

And

"But I don't keep up with the literature, so maybe I'm just way behind
everyone else."

You put an awful lot of trust in someone Rick, who doesn't seem to care all
that much about his current level of knowledge, or his willingness to look
at new ideas or new advances when they happen.

I guess one day an over active student will discover exactly what Powers has
spent so much time trying to avoid and be able to solidify the concept of
control in human behavior.

Because Rick, as much as you and Bill want to deny it, Human behavior is a
_great_ deal more than a simple tracking task and the questions people have
about control are not about how a six legged caterpillar moves on the
ground, it's about how control affects the cognitive processes we use on a
daily basis for our own choices and choices that ultimately affect others.

You can pooh-pooh this all you want, but I recently realized why George
Richardson felt you (meaning Bill P)'went-into-the tank' with your tracking
task. George actually thinks the world of you and your work and he was very
disappointed that you did not go after the 'cognitive' or higher level stuff
rather than the mundane lower level stuff.

Your refusal to address the questions posed by Bruce Gregory and others and
more importantly your insistence on working from the neurons on up is an
absolute disaster.

Until PCT can begin to model an integrated set of processes that account for
our cognitive behavior, PCT will be a fringe, incomplete, view of human
behavior at best.

You don't make progress by sticking your head in the sand and making believe
other things don't exist.

One day you are going to find out that there is no status quo to protect

Marc

13

[From Rick Marken (2004.09.17.2310)]

Marc Abrams (2004.09.17.2305)--

What a crock. Neither one of you is man enough to admit that you might
have
made a mistake.

I made a mistake in translating Bill's form for the derivative into a
form similar to yours. Now why don't you admit that you made a mistake
when you posted your version of the equation for the derivative. The
equation you posted was:

             Lim f(x + h) - f(x)
f'(x) = x ->0 ---------------
                                    h ;Where h
is very small change in x

You say you copied this out of your calculus book. Are you sure you
copied it correctly or might you have made a mistake?

RSM

···

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Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400

14

From [Marc Abrams (2004.09.18.0909)]

[From Rick Marken (2004.09.17.2310)]

You say you copied this out of your calculus book. Are you sure you
copied it correctly or might you have made a mistake?

Not a chance in hell. In some texts the h is replaced by a triangle and x,
indicating a very small _change_ in x, but the difference quotient is the
basis for differentiation, and that is what you saw.

How about my other questions? You want a 'discussion', lets have one.

What is the basis you use for accepting the sample of 15 quarters of data as
being representative of the entire _population_, that is, of _all_ quarters
of data? Any reasonable student in statistics 101 would ask you this
question. And by what set of correlations do you make the pronouncement that
'growth' in the GNP represents what you say it does?

Given the fact that correlation does _not_ indicate _causality_, what
additional analysis have you done to back up your claims?

Marc

15

[From Bill Powers (2004.09.18.0920 MDT)]

Rick Marken (2004.09.17.0945) --

A loose end:

Growth rate (the black trace) is computed in the standard way:
[GNP(t)/GNP(t-1)]-1.

That is the same as

GNP(t) - GNP(t-1)

···

------------------
    GNP(t-1)

which converges to the "logarithmic derivative" -- the derivative as a
fraction of the total value. The derivative of log(u) is(1/u)du, hence the
name. It has the advantage that the same fractional change will have the
same value regardless of the value of GNP.

Nice catch about letting x instead of h go to zero. Goes to show what
happens when you copy formulas without understanding them -- or when you
make a slip.

Best,

Bill P.

16

[From Rick Marken (2004.09.18.0840)]

Marc Abrams (2004.09.18.0909)]

[From Rick Marken (2004.09.17.2310)]

You say you copied this out of your calculus book. Are you sure you
copied it correctly or might you have made a mistake?

Not a chance in hell.

OK. So you're sticking with this definition of the derivative:

             Lim f(x + h) - f(x)
f'(x) = x ->0 ---------------
                                 h ;Where h is
very small change in x

This formula seems strange to me because x is the argument of the
function f(). The derivative of f(x) -- f'(x) or y in Bill's dy/dx
notation -- as I understand it, is the rate of change in f(x) at x. So
defining f'(x) as the limit of the ratio (f(x+h) - f(x))/h as x-> 0
seems like nonsense. The rate of change in f(x) is given by the
formula, (f(x+h) - f(x))/h. This ratio says how much f(x) changes per h
amount of change in x. I think the idea of a derivative is that it
represents how much f(x) changes at a particular value of x. That is,
the derivative of f(x) is the amount of change in f(x) per
infinitesimally small change in x, dx, that is when h, the size of the
change in x, approaches 0. In my calculus book, the definition of the
derivative of f(x) is, thus,

f'(x) = lim f(x+h) - f(x)
           h->0 ----------------
                              h
It is h, not x, that approaches 0.

What is the basis you use for accepting the sample of 15 quarters of
data as
being representative of the entire _population_, that is, of _all_
quarters
of data?

I don't understand the question. What 15 quarters was I using as a
sample to estimate a population of quarters?

And by what set of correlations do you make the pronouncement that
'growth' in the GNP represents what you say it does?

Could you frame your questions in terms of the data I presented,
please. What correlations? What pronouncement about growth in GNP
representing what? What did I say growth in GNP represents other than
growth in GNP?

Given the fact that correlation does _not_ indicate _causality_, what
additional analysis have you done to back up your claims?

Now are you talking about the lagged correlations between growth and
investment? If so, I don't believe I made any claims about causality.
The correlations describe the degree of linear relationship between
variables. They are statistically significant (I reported the results
of significance tests in terms of confidence intervals around the
correlations) which simply means that, given the number of samples on
which they are based, the "true" correlations are likely to be close to
the values of those computed.

RSM

···

---
Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400

17

[From Bruce Abbott (2004.09.18.1210 EST)]

Rick Marken (2004.09.18.0840) --

I don't understand the question. What 15 quarters was I using as a
sample to estimate a population of quarters?

<snip>

The correlations describe the degree of linear relationship between
variables. They are statistically significant (I reported the results
of significance tests in terms of confidence intervals around the
correlations) which simply means that, given the number of samples on
which they are based, the "true" correlations are likely to be close to
the values of those computed.

Just a point about your statistical analysis. Your significance tests have
meaning if, and only if, the 15 quarters of data you have represent a
sample from a larger population of quarters, and the sample is taken
randomly from that population. In the first paragraph above you appear to
be implying that your data are the entire population of quarters of
interest. And even if they are a sample, they do not appear to be a random
sample. If this is a population, then the observed correlation is what it
is. You don't need a significance test to tell you that it isn't likely to
be zero.

Bruce A.

18

From [Marc Abrams (2004.09.18.1308)]

[From Rick Marken (2004.09.18.0840)]

In my calculus book, the definition of the

derivative of f(x) is, thus,

f'(x) = lim f(x+h) - f(x)
           h->0 ----------------
                              h
It is h, not x, that approaches 0.

From Bill Powers [From Bill Powers (2004.09.18.0920 MDT)]

Nice catch about letting x instead of h go to zero. Goes to show what
happens when you copy formulas without understanding them -- or when you
make a slip.

···

------------
What a tag team. Boy did you get me but good Rick. You sure showed me

Forget about the fact that h _never_ gets to zero because _LIMITS_ _NEVER_
do, but quotients can, like Powers thinks they can.

Yes, I made a transcription mistake, because I used h, rather that then the
triangle and x to indicate a small change in x, but that still doesn't
change the fact that my equation was the right one and Powers was wrong. Get
over it.

> What is the basis you use for accepting the sample of 15 quarters of
> data as
> being representative of the entire _population_, that is, of _all_
> quarters
> of data?

I don't understand the question. What 15 quarters was I using as a
sample to estimate a population of quarters?

Was it a _random_ sample? Or do you not know what a random sample is?
You wrote a book on statistical methods, do I need to spell it out for you?

> And by what set of correlations do you make the pronouncement that
> 'growth' in the GNP represents what you say it does?

Could you frame your questions in terms of the data I presented,
please. What correlations? What pronouncement about growth in GNP
representing what? What did I say growth in GNP represents other than
growth in GNP?

No, you equated the deficit and debt as being 'bad' and the 'surplus' as
being 'good'. I want to know why responsible debt is a 'bad' thing. Deficits
are intolerable but debts are indeed needed.

What is, and what was, the purpose of showing the amount of growth in the
GNP?

Are you really attributing the 'growth' in the economy to a President? Or an
administration? If so, please back up these claims. I assume you are using
the 15 quarters for this argument. Give me a break.

Now are you talking about the lagged correlations between growth and
investment? If so, I don't believe I made any claims about causality.
The correlations describe the degree of linear relationship between
variables. They are statistically significant (I reported the results
of significance tests in terms of confidence intervals around the
correlations) which simply means that, given the number of samples on
which they are based, the "true" correlations are likely to be close to
the values of those computed.

NO. I am asking for the basis of your _opinion_, on the significance of an
aspect of the economy, you attribute to the President or administration.

Marc

19

[From Bill Powers (2004.09.18.1609 MDT)]

Rick Marken (2004.09.18.0840) --

[To Marc A]: OK. So you're sticking with this definition of the derivative:

             Lim (x + h) - f(x)
f'(x) = x ->0 ---------------;
                        h
where h is a very small change in x

This is a simple miscopying of x for h. As you say it is h that approaches
zero, not x. As written above, the definition of the derivative is
incorrect. If the error is acknowledged, fine. People make mistakes. If
mistakes are not admitted when pointed out, further discussion is pointless.

Best,

Bill P.

20

From [Marc Abrams (2004.09.18.1826)]

[From Bill Powers (2004.09.18.1609 MDT)]
>[To Marc A]: OK. So you're sticking with this definition of the
derivative:
>
>> Lim (x + h) - f(x)
>> f'(x) = x ->0 ---------------;
>> h
>> where h is a very small change in x

This is a simple miscopying of x for h. As you say it is h that approaches
zero, not x. As written above, the definition of the derivative is
incorrect. If the error is acknowledged, fine. People make mistakes. If
mistakes are not admitted when pointed out, further discussion is
pointless.

First, thank you for responding like a human being. I appreciate it and
thank you for it. I agree with you which is why I said; and apparently you
missed this post earlier in the day in response to Rick,

From [Marc Abrams (2004.09.18.1308)]

What a tag team. Boy did you get me but good Rick. You sure showed me

Forget about the fact that h _never_ gets to zero because _LIMITS_ _NEVER_
do, but quotients can, like Powers thinks they can.

Yes, I made a transcription mistake, because I used h, rather then the
triangle and x to indicate a small change in x, but that still doesn't
change the fact that my equation was the right one and Powers was wrong. Get
over it.

···

-----------------------

Lets all move on. It's over.

Marc