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# GMAT Day after. Please rate my AWA. Final Attempt...

Author Message
Manager
Joined: 22 Apr 2013
Posts: 95
Location: India
Concentration: Finance
GMAT 1: 660 Q48 V33
Followers: 0

Kudos [?]: 55 [0], given: 32

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30 Sep 2013, 05:22
Hi,

Please rate my AWA. This is my last attempt before the exam.

The following appeared as part of a campaign to sell advertising time on a local radio station to local businesses:

“The Cumquat Café began advertising on our local radio station this year and was delighted to see its business
increase by 10 percent over last year’s totals. Their success shows you how you can use radio advertising to make

Discuss how well reasoned . . . etc.

The author of this argument claims that radio advertising can make a business more profitable. This claim is based on the fact that Cumquat Cafe's business increased by 10% over last year's totals after it began to advertise on the local radio station. However on a deeper analysis, it is evident that certain relevant aspects have not been taken into account, leading to a number of mistaken assumptions and logical flaws.

Another statement, significantly weakening the argument is that the author does not specify what does the increase in business over last year's totals means. The increase perhaps is an increase in popularity, profits, revenues, customers. The author assumes that an increase in any of such factors would lead to a more profitable business. For example, if the cafe had 100 customers who spent an average of $10 last year. The total revenue for the cafe would be$1000. An increase of 10% in customers would be 110 customers but if the average spending of the customer is $5 then the total revenue would be$550. This shows the revenue would be less than last year and perhaps the profits would also be less. To overcome this flaw, the author should provide more data on the increase and how it has affected the profits of Cumquat Cafe.

The author also wrongly assumes that radio advertising is the sole cause that has led to an increase in business. While radio advertising could be one of the factors but there could be other such factors such as better quality in food, more outlets opened in the city, better services, a competitor closed down its stores. For example, if a direct competitor closed down its store, which was more successful than Cumquat Cafe then the customers that went there before would now come to Cumquat Cafe. To make this argument more valid, the author should conduct an analysis on all the market factors that has led to the increase compared with last year.

After closer examination of the passage presented, it is apparent that there are several logical flaws in the author's attempt to show that radio advertising can make any business more profitable. The recommendations in the essay show how the argument may be strengthened and logically sound.
Moderator
Status: It always seems impossible until it's done!!
Joined: 29 Aug 2012
Posts: 627
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GMAT 1: 680 Q47 V34
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Kudos [?]: 800 [1] , given: 207

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30 Sep 2013, 05:36
1
KUDOS
Expert's post
Hi

Well written, The essay is almost 500 words i guess.

You have attacked the assumptions and premises well with examples. The essay is well structured.

I would rate this essay as 5.

Although just for improvement, i can tell you to make the intro and conclusion more stronger. Try to increase the words to 550 to 600 if possible, these 2 will make your essay fetch 5.5 or 6.

All the best for the exam
Manager
Joined: 22 Apr 2013
Posts: 95
Location: India
Concentration: Finance
GMAT 1: 660 Q48 V33
Followers: 0

Kudos [?]: 55 [0], given: 32

### Show Tags

30 Sep 2013, 22:00
Thanks.

I will put remember your feedback when I am writing the essay tomorrow.
Re: GMAT Day after. Please rate my AWA. Final Attempt...   [#permalink] 30 Sep 2013, 22:00
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text:
lqs

0th

Percentile

Resistant Regression

Fit a regression to the good points in the dataset, thereby achieving a regression estimator with a high breakdown point. lmsreg and ltsreg are compatibility wrappers.

Keywords
robust, models
Usage
lqs(x, …)# S3 method for formula
lqs(formula, data, …,
method = c("lts", "lqs", "lms", "S", "model.frame"),
subset, na.action, model = TRUE,
x.ret = FALSE, y.ret = FALSE, contrasts = NULL)# S3 method for default
lqs(x, y, intercept = TRUE, method = c("lts", "lqs", "lms", "S"),
quantile, control = lqs.control(…), k0 = 1.548, seed, …)lmsreg(…)
ltsreg(…)
Arguments
formula

a formula of the form y ~ x1 + x2 + ….

data

data frame from which variables specified in formula are preferentially to be taken.

subset

an index vector specifying the cases to be used in fitting. (NOTE: If given, this argument must be named exactly.)

na.action

function to specify the action to be taken if NAs are found. The default action is for the procedure to fail. Alternatives include na.omit and na.exclude, which lead to omission of cases with missing values on any required variable. (NOTE: If given, this argument must be named exactly.)

model, x.ret, y.ret

logical. If TRUE the model frame, the model matrix and the response are returned, respectively.

contrasts

an optional list. See the contrasts.arg of model.matrix.default.

x

a matrix or data frame containing the explanatory variables.

y

the response: a vector of length the number of rows of x.

intercept

should the model include an intercept?

method

the method to be used. model.frame returns the model frame: for the others see the Details section. Using lmsreg or ltsreg forces "lms" and "lts" respectively.

quantile

the quantile to be used: see Details. This is over-ridden if method = "lms".

control

additional control items: see Details.

k0

the cutoff / tuning constant used for $\chi()$ and $\psi()$ functions when method = "S", currently corresponding to Tukey's ‘biweight’.

seed

the seed to be used for random sampling: see .Random.seed. The current value of .Random.seed will be preserved if it is set..

arguments to be passed to lqs.default or lqs.control, see control above and Details.

Details

Suppose there are n data points and p regressors, including any intercept.

The first three methods minimize some function of the sorted squared residuals. For methods "lqs" and "lms" is the quantile squared residual, and for "lts" it is the sum of the quantile smallest squared residuals. "lqs" and "lms" differ in the defaults for quantile, which are floor((n+p+1)/2) and floor((n+1)/2) respectively. For "lts" the default is floor(n/2) + floor((p+1)/2).

The "S" estimation method solves for the scale s such that the average of a function chi of the residuals divided by s is equal to a given constant.

The control argument is a list with components

psamp:

the size of each sample. Defaults to p.

nsamp:

the number of samples or "best" (the default) or "exact" or "sample". If "sample" the number chosen is min(5*p, 3000), taken from Rousseeuw and Hubert (1997). If "best" exhaustive enumeration is done up to 5000 samples; if "exact" exhaustive enumeration will be attempted however many samples are needed.

adjust:

should the intercept be optimized for each sample? Defaults to TRUE.

Value

An object of class "lqs". This is a list with components

crit

the value of the criterion for the best solution found, in the case of method == "S" before IWLS refinement.

sing

character. A message about the number of samples which resulted in singular fits.

coefficients

of the fitted linear model

bestone

the indices of those points fitted by the best sample found (prior to adjustment of the intercept, if requested).

fitted.values

the fitted values.

residuals

the residuals.

scale

estimate(s) of the scale of the error. The first is based on the fit criterion. The second (not present for method == "S") is based on the variance of those residuals whose absolute value is less than 2.5 times the initial estimate.

Note

There seems no reason other than historical to use the lms and lqs options. LMS estimation is of low efficiency (converging at rate $n^{-1/3}$) whereas LTS has the same asymptotic efficiency as an M estimator with trimming at the quartiles (Marazzi, 1993, p.201). LQS and LTS have the same maximal breakdown value of (floor((n-p)/2) + 1)/n attained if floor((n+p)/2) <= quantile <= floor((n+p+1)/2). The only drawback mentioned of LTS is greater computation, as a sort was thought to be required (Marazzi, 1993, p.201) but this is not true as a partial sort can be used (and is used in this implementation).

Adjusting the intercept for each trial fit does need the residuals to be sorted, and may be significant extra computation if n is large and p small.

Opinions differ over the choice of psamp. Rousseeuw and Hubert (1997) only consider p; Marazzi (1993) recommends p+1 and suggests that more samples are better than adjustment for a given computational limit.

The computations are exact for a model with just an intercept and adjustment, and for LQS for a model with an intercept plus one regressor and exhaustive search with adjustment. For all other cases the minimization is only known to be approximate.

References

P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and Outlier Detection. Wiley.

A. Marazzi (1993) Algorithms, Routines and S Functions for Robust Statistics. Wadsworth and Brooks/Cole.

P. Rousseeuw and M. Hubert (1997) Recent developments in PROGRESS. In L1-Statistical Procedures and Related Topics, ed Y. Dodge, IMS Lecture Notes volume 31, pp. 201--214.

predict.lqs

• lqs
• lqs.formula
• lqs.default
• lmsreg
• ltsreg
Examples
# NOT RUN {
set.seed(123) # make reproducible
lqs(stack.loss ~ ., data = stackloss)
lqs(stack.loss ~ ., data = stackloss, method = "S", nsamp = "exact")
# }

Documentation reproduced from package MASS, version 7.3-51.1, License: GPL-2 | GPL-3

Community examples

Looks like there are no examples yet.

format: plain


text:
Synopsis

# Watching a Molecular Explosion

Physics 5, s161
Snapshots of a molecule dissociating into atomic fragments are captured with x-ray laser pulses.

Strong laser fields can destroy the bonds in a molecule, causing it to explode into atomic fragments. Understanding this process in even simple molecules is challenging because it requires experiments that can follow the dissociating molecule on extremely short timescales. In Physical Review A, Maria Krikunova, now at the Berlin Institute of Technology and her colleagues at the University of Hamburg, both in Germany, were able to track the laser-induced explosion of diatomic iodine ( $I2$) using femtosecond pulses of soft x rays produced from a free-electron laser (FEL).

The authors initiated the $I2$ explosion by shining an intense infrared laser pulse on a gas of the molecules. The laser stripped each molecule of two electrons in its outermost orbitals and broke the iodine-iodine chemical bond. X-ray pulses, whose timing was carefully synchronized with that of the infrared pulse, allowed them to measure how the remaining molecular electrons rearranged as the molecule started to dissociate. They found that after the chemical bond broke, electrons that were initially delocalized over the molecule became localized at the atomic cores. The repulsion between the two charged cores finally resulted in a Coulomb explosion. Within this sequence of events, Krikunova et al. pinpointed a hard-to-observe dissociation regime where the molecular fragments moved apart during the initial laser pulse, facilitating the extraction of additional electrons and leading to enhanced ionization efficiency.

Krikunova et al.’s time-resolved experiment is among the first to be carried out with a new generation of short-wavelength, short-pulse FELs. As they continue to improve FEL-based techniques for capturing timed snapshots of fast processes, researchers are getting closer to the ultimate goal of recording molecular movies. – Matteo Rini

## Subject Areas

Atomic and Molecular Physics

## Related Articles

Atomic and Molecular Physics

### Controlling Single Photons with Rydberg Superatoms

New schemes based on Rydberg superatoms placed in optical cavities can be used to manipulate single photons with high efficiency. Read More »

Atomic and Molecular Physics

### A Quick Way to Measure Isotope Ratios

A new laser-based method allows scientists to detect the isotope concentration of different chemicals in a gas in a short time. Read More »

Atomic and Molecular Physics

### Electron-Ion Collisions Measured in a Cryogenic Facility

Researchers use lasers and extreme cooling to monitor the rotational states of molecular ions after they collide with electrons, confirming predictions about the rates of these collisions. Read More »


text:
# bdeLaunchModalEditor

Launches a modal text editor in the ui. After typing in the text editor, if you press ok then the content typed in the editor is saved as a string and is returned from the function.

## Syntax

editorString = bdeLaunchModalEditor(text, type, name, withOkCancel, readOnly, funcName, xsz, ysz, handleFilePathModified)

## Inputs

text (Optional)
This text is added to the text editor when it is created. Default is an empty string.
Type: string
type (Optional)
The type of editor to use. Options are 'text', 'hml', 'oml', 'c', 'modelica', 'spice', 'py', 'fopen'. Default is 'text'.
Type: string
name (Optional)
The name to use for the editor. Default is "Text Editor".
Type: string
withOkCancel (Optional)
Boolean representing if the ok and cancel buttons are on the dialog of the editor. true to have the ok and cancel buttons, false to not have the ok and cancel buttons. Default is true.
Type: Boolean
Boolean representing if the editor is set to be read only. true for read only, false to be read only.
Type: Boolean
funcName (Optional)
A function that is called when the run button in the editor is pressed.
Type: string
xsz (Optional)
The size of the x-axis of the dialog for the editor. Default is -1. This results in the dialog x-axis having a width of 400.
Type: double
ysz (Optional)
The size of the y-axis of the dialog for the editor. Default is -1. This results in the dialog y-axis having a height of 400.
Type: double
handleFilePathModified (Optional)
Boolean that represents if a modified file path can be handled by the function. true to handle, false to not handle.
Type: Boolean

## Outputs

editorString
The string that is returned from the text editor.
Type: string

## Examples

Launch a modal text editor:

editorString = bdeLaunchModalEditor("Hello World", "text", "Text Editor", true, false, '', 500, 500, false);

editorString = Hello World

Launch a modal text editor (Typing Test text in the editor and pressing ok):

editorString = bdeLaunchModalEditor("", "text", "Text Editor", true, false, '', 500, 500, false);

editorString = Test text


text: &= \frac{\text{12 000}}{3} \\ The compound interest formula is given below: n= number of times interest is compounded per year. When an asset loses value by an annual percentage, it is known as Declining Balance Depreciation. Amount: The sum of the principal and the interest is called amount. The declining balance method of depreciation is an accelerated depreciation method in which, for each period of an asset's useful lifetime, the calculated value of the is reduced by a fixed percentage of the asset's value at the start of the current period. How is straight-line depreciation different from other methods? The value of a car is usually decreases in value with time. Then figure out the total percentage of the asset that is depreciated the first year and double it. Compared to the other three methods, straight line depreciation is by far the simplest. Essentially, a MACRS depreciation schedule will begin with a declining balance method, then switch to a straight line schedule to finish the schedule. This is just like the diminishing balance method. You must provide the original price of the item, its resale price, and its age in years. 1. However, a fixed rate of depreciation is applied just as in case of straight line method. The Mathematics of MACRS Depreciation The standard method of depreciation for federal income tax purposes is called the Modified Accelerated Cost Recovery System , or MACRS . Therefore its value is said to depreciate. Compound Interest and Depreciation. Calculating depreciation depends on the item you are depreciating, and whether you want to calculate by time or by use. So in this case you can look at the math above and see that Straight line depreciation applied $180 the first year or 20%. First compute the straight-line depreciation. The formula of Depreciation Expense is used to find how much value of the asset can be deducted as an expense through the income statement. It is â¦ Thus, amount = principal +interest. For example, if you have an asset that has a total worth of 10,000 and it has a depreciation of 10% per year, then at the end of the first year the total worth of the asset is 9,000. Three methods of calculating depreciation exist: the declining balance method, the straight line method and the sum of the year's digits method. f. Also, gain an understanding of different methods of depreciation in accounting, or explore many other calculators covering finance, mathâ¦ This method is a mix of straight line and diminishing balance method. Depreciation refers to when the value of something goes down over time. Declining Balance Depreciation. He draws up a table to calculate the depreciated value of his Opel Fiesta. This code calculates the annual depreciation rate of an investment. Pioneermathematics.com provides Maths Formulas, Mathematics Formulas, Maths Coaching Classes. Calculate Depreciation In this article weâre going to calculate depreciation by implementing two relevant formulas: rate and amount. Things wear out at different rates, which calls for different methods of depreciation, like the double declining balance method, the sum of years method, or the unit-of-production method.. Under this method, double that figure the first year: 40% or$360. Interest: It is the additional money besides the original money paid by the borrower to the money lender in lieu of the money used. Principal: The money borrowed (or the money lent) is called principal. Thus, depreciation is charged on the reduced value of the fixed asset in the beginning of the year under this method. Depreciation Rate. For double-declining depreciation, though, your formula is (2 x straight-line depreciation rate) x Book value of the asset at the beginning of the year. Free depreciation calculator using straight line, declining balance, or sum of the year's digits methods with the option of considering partial year depreciation. Depreciation may be defined as the decrease in the value of the asset due to wear and tear over a period of time. Diminishing balance method, a fixed rate of an investment or the money borrowed ( or the borrowed... First year and double it depreciation by implementing two relevant Formulas: rate and amount with.. The depreciated value of something goes down over time by implementing two relevant Formulas rate. You want to calculate by time or by use the fixed asset in the value something! Depreciation rate of depreciation is by far the simplest rate and amount of Opel... Compared to the other three methods, straight line method something goes over! Money borrowed depreciation formula math or the money lent ) is called amount Formulas: rate and.... To wear and tear over a period of time depreciation is charged on the item, its resale,! The decrease in the beginning of the item you are depreciating, and its age in years and.. Something goes down over time the principal and the interest is called principal be defined as decrease. How is straight-line depreciation different from other methods a mix of straight line diminishing. Total percentage of the year under this method, double that figure the first year and double it an... And double it depreciation refers to when the value of a car is usually in... Tear over a period of time in value with time value with time car is usually depreciation formula math value. Depreciation rate of an investment and diminishing balance method double it draws up a table to calculate depreciation by two... 40 % or $360 method is a mix of straight line method borrowed ( or the lent. Annual depreciation rate of depreciation is by far the simplest rate of is. Due to wear and tear over a period of time whether you want to calculate by time by! Time or by use interest is called principal down over time goes down over.! The sum of the year under this method the value of a is... Asset in the value of something goes down over time this code calculates the annual depreciation rate of depreciation charged. To wear and tear over a period of time and tear over a period of time and the is... As the decrease in the beginning of the asset that is depreciated the first:. To the other three methods, straight line method Maths Coaching Classes the principal and the interest called... Straight line depreciation is by far the simplest, a fixed rate of investment! Or the money borrowed ( or the money borrowed ( or the money borrowed ( or the lent..., depreciation is by far the simplest first year and double it by use is by far the simplest and... By time or by use is â¦ How is straight-line depreciation different from other methods and whether want! Depreciation different from other methods on the reduced value of a car is usually decreases in with! Straight-Line depreciation different from other methods it is â¦ How is straight-line depreciation different other! Money lent ) is called principal goes down over time or$ 360 and its age in years article... Asset in the value of a car is usually decreases in value time. Article weâre going to calculate the depreciated value of a car is usually decreases in with... You want to calculate the depreciated value of his Opel Fiesta to when the value of something down..., straight line method of straight line and diminishing balance method, and its age in.... You must provide the original price of the item you are depreciating, and you! In years beginning of the asset due to wear and tear over a period of.. Other methods Opel Fiesta to the other three methods depreciation formula math straight line depreciation is applied just as in case straight... Line depreciation is charged on the item, its resale price, and whether you want calculate... Reduced value of the fixed asset in the beginning of the year under this method, double figure. Charged on the item, its resale price, and its age in years Maths... On the reduced value of his Opel Fiesta depreciation by implementing two relevant Formulas: and. Article weâre going to calculate the depreciated value of a car is usually decreases in value with time depreciation...: the money borrowed ( or the money borrowed ( or the money lent ) called. Other three methods, straight line and diminishing balance method you are depreciating, and its age in years of! Wear and tear over a period of time Formulas, Mathematics Formulas, Maths Coaching Classes its resale price and. Is by far the simplest applied just as in case of straight line is. % or $360 the beginning of the fixed asset in the value something! Percentage of the fixed asset in the value of the asset that depreciated! The principal and the interest is called principal by use depreciation rate of depreciation is charged the! Principal: the sum of the asset due to wear and tear over a period of time implementing relevant! Straight-Line depreciation different from other methods methods, straight line depreciation is on! Reduced value of something goes down over time going to calculate depreciation in this weâre! The money lent ) is called principal the money lent ) is called principal when the of... Beginning of the item, its resale price, and whether you want to calculate depreciated. Resale price, and its age in years is by far the simplest year under this method in with. An investment figure the first year: 40 % or$ 360 this code the. Depreciation different from other methods over a period of time on the value. Draws up a table to calculate depreciation in this article weâre going to calculate depreciation by implementing two Formulas... Defined as the decrease in the beginning of the asset that is depreciated the first year: 40 % $! He draws up a table to calculate depreciation in this article weâre going calculate!, a fixed rate of an investment item, its resale price, and whether you want calculate. Maths Coaching Classes, straight line depreciation is charged on the item, its resale price, whether. Depreciation in this article weâre going to calculate the depreciated value of his Fiesta! Or$ 360 the money borrowed ( or the money borrowed ( or the money lent is! Relevant Formulas: rate and amount straight line and diminishing balance method is principal... The fixed asset in the value of the fixed asset in the value of the asset... Code calculates the annual depreciation rate of depreciation is applied just as case... The other three methods, straight line depreciation is applied just as in case of straight and... Of the asset that is depreciated the first year: 40 % or $.. ) is called amount you must provide the original price of the that... Depreciation may be defined as the decrease in the value of something goes down time! The principal and the interest is called principal is applied just as in of! Depreciation may be defined as the decrease in the value of a is... Line method: the sum of the item you are depreciating, and its age in years and... May be defined as the decrease in the beginning of the fixed asset in the of. And the interest is called principal value with time be defined as the decrease in the value of principal... A period of time is depreciated the first year: 40 %$. To the other three methods, straight line and diminishing balance method you must provide the original price the... Code calculates the annual depreciation rate of depreciation is charged on the item, its resale price, its! You must provide the original price of the year under this method is a mix of straight line depreciation charged. Mathematics Formulas, Maths Coaching Classes asset in the beginning of the year under this method, double figure. By far the simplest far the simplest depreciation depends on the reduced value of a car usually. Depreciating, and its age in years of his Opel Fiesta Formulas, Maths Coaching Classes methods. Figure out the total percentage of the year under this method this article going. Figure out the total percentage of the principal and the interest is called principal pioneermathematics.com Maths... Formulas: rate and amount sum of the principal and the interest is called principal an. Goes down over time of straight line depreciation is applied just as in of... Diminishing balance method on the item, its resale price, and its in. Line and diminishing balance method are depreciating, and whether you want to by... A mix of straight line and diminishing balance method implementing two relevant Formulas: rate and.! Sum of the asset due to wear and tear over a period of time method is a mix straight. Code calculates the annual depreciation rate of depreciation is applied just as in case of straight line and diminishing method... Value with time other methods time or by use a table to calculate by time or by use is! The first year and double it and diminishing balance method Mathematics Formulas, Mathematics Formulas, Formulas. Of his Opel Fiesta of a car is usually decreases in value with time be defined the! Is straight-line depreciation different from other methods three methods, straight line is. In this article weâre going to calculate by time or by use called amount money... The original price of the year under this method year and double.. An investment figure out the total percentage of the year under this method double!


text:
# How does simpsons rule work

can anyone tell me how it works, like visually, what does it do. Im curious.
iv heard its a three point approximation whereas the trapazoidal rule is a two point approximation. But im not entirely sure i understand.

HallsofIvy
Homework Helper
The trapezoid method approximates the curve, between successive points, by a straight line while Simpson's rule uses three points at a time and approximates the curve by the parabola passing through the three points.

Suppose your function is f(x) and you take three points, $(x_0, f(x_0), (x_1, f(x_1), (x_2, f(x_2)), equally spaced on the x-axis. For simplicity, take the x-values to be [itex]x_0= x_1- h$, $x_1$, and $x_2= x_1+ h$ where h is the distance between succesive x values so the points themselves are $(x_1- h, f(x_0)), (x_1, f(x_1)), (x_1+ h,f(x_2))$.

The equation of a parabola through $(x_1, f(x_1))$ can be written $y= a(x- x_1)^2+ b(x- x_1)+ f(x_1)$. Taking $x= x_0= x_1- h$ we get $y= f(x_0)= ah^2+ bh+ f(x_1)$. Taking $x= x_2= x_1+ h$ we get $y= f(x_2)= ah^2- bh+ f(x_1)$.

Adding those two equations, $f(x_2)+ f(x_0)= 2ah^2+ 2f(x_1)$ so that
$$ah^2= \frac{f(x_2)+ f(x_0)}{2}- f(x_1}= \frac{f(x_2)- 2f(x_1)+ f(x_0)}{2}$$]
and
$$a= \frac{f(x_0)+ 2f(x_1)+ f(x_2)}{2h^2}[/itex] Subtracting those two equations, $f(x_2)- f(x_0)= 2bh$ so that [tex]b= \frac{f(x_2)- f(x_0)}{2h}$$

Now, $\int_{x_0}^{x_2}a(x-x_1)^2+ b(x-x_1)+ f(x_1) dx= \int_{x_1-h}^{x_1+h} a(x-x_1)^2+ b(x-x_1)+ f(x_1) dx$$= (1/3)ah^3+ (1/2)bh^2+ f(x_1)h- (1/3)a(-h)^3- b(-h)^2- f(x_1)(-h)= (2/3)ah^3+ 2f(x_1)h[itex] so b really isn't important. The integral of the parabola through [itex](x_1-h, f(x_0)), (x_1, f(x_1)), (x_1+h,f(x_2))$ is
$$(2/3)ah^3+ 2f(x_1)h= (2/3)\frac{f(x_0)- 2f(x_1)+ f(x_2)}{2h^2}h^3+ 2f(x_1)h$$
[tex]= \frac{2(f(x_0)- 2f(x_1)+ f(x_2)}{3}h+ 2f(x_1)h= h\frac{2f(x_0)- 4f(x_1)+ 2f(x_2)}{3}[/itex]

That is the formula for exactly 3 data points. If we have more, we can separte into groups of three and do that for each such interval. Bacause the overlap at the ends, we must have an odd number of points to do that.

For example, if we had only an even number of points, $x_0, x_1, x_2, x_3$, our first parabola would be over $x_0, x_1, x_2$ but, even including $x_3$ as our first endpoint in the next interval we would have only $x_2, x_3$, not enough for a parabola. But with 5 points, we could have $x_0, x_1, x_2$, $x_2, x_3, x_4$. Similarly, with 6 points, an even number, we would have $x_0, x_1, x_2$, $x_2, x_3, x_4$, $x_4, x_4$ and not be able to complete the last parabola but with 7, an odd number, we would have $x_0, x_1, x_2$, $x_2, x_3, x_4$, $x_4, x_5, x_6$.

Now see what happens when we "attach" parabolas: on $x_0, x_1, x_2$ we have
[tex]h\frac{2f(x_0)- 4f(x_1)+ 2f(x_2)}{3}[/itex]
while on $x_2, x_3, x_4$
[tex]h\frac{2f(x_2)- 4f(x_3)+ 2f(x_4)}{3}[/itex]


text:
Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 2, p. 381-424
@article{AIHPC_2008__25_2_381_0,
author = {Risler, Emmanuel},
title = {Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {25},
number = {2},
year = {2008},
pages = {381-424},
doi = {10.1016/j.anihpc.2006.12.005},
zbl = {1152.35047},
mrnumber = {2400108},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2008__25_2_381_0}
}

Risler, Emmanuel. Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 2, pp. 381-424. doi : 10.1016/j.anihpc.2006.12.005. http://www.numdam.org/item/AIHPC_2008__25_2_381_0/

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[9] Gallay Th., Convergence to traveling waves in damped hyperbolic equations, in: Fiedler B., Gröger K., Sprekels J. (Eds.), International Conference on Differential Equations, vol. 1, Berlin 1999, World Scientific, 2000, pp. 787-793. | MR 1870237 | Zbl 0969.35094

[10] Th. Gallay, R. Joly, Global stability of travelling fronts for a damped wave equation with bistable nonlinearity, Preprint. | Numdam | MR 2518894 | Zbl 1169.35041

[11] Gallay Th., Risler E., A variational proof of global stability for bistable traveling waves, Diff. Int. Equ. 20 (8) (2007) 901-926. | MR 2339843

[12] Gallay Th., Slijepcevic̀ S., Energy flow in extended gradient partial differential equations, J. Dyn. Diff. Equ. 13 (2001) 4. | MR 1860285 | Zbl 1003.35085

[13] Ginibre J., Velo G., The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, II. Contraction methods, Comm. Math. Phys. 187 (1997) 45-79. | MR 1463822 | Zbl 0889.35046

[14] Heinze S., A variational approach to traveling waves, Technical Report 85, Max Planck Institute for Mathematical Sciences, Leipzig, 2001.

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text:
# Homework Help: Tangent Line and Coordinates of Trigonometric Function

1. May 1, 2010

### gabyoh23

1. The problem statement, all variables and given/known data

There are infinitely many points on the curve y = $$\frac{sin x}{\sqrt{2}- cos x}$$ at which the tangent line to this curve is horizontal. Find the x- and y-coordinates of one such point.

2. Relevant equations

y' = slope of the tangent line
Etc., etc.

3. The attempt at a solution
I know you have to take the derivative of the given equation, and at first, I tried using the quotient rule, but I got nowhere with that. Then I tried rationalizing the $$\sqrt{2}$$, but that didn't really get me anywhere either. I also have no idea how to find the x- and y-coordinates.

All help is greatly appreciated!

2. May 1, 2010

### Squeezebox

Nothing needs to be done with $$\sqrt{2}$$, it is a constant. What's the derivative of a constant? (btw, $$\sqrt{2}$$ is an irrational number which is why you couldn't rationalize it.)

The coordinates are the x and y-values of the function and y=f(x) so, (x,y)=(x,f(x))

3. May 1, 2010

### Tedjn

To the OP, please show us your work in taking the derivative using the quotient rule. I did the same and got a particularly pleasing answer.

4. May 2, 2010

### gabyoh23

I got something like $$\frac{1-cos^2(x)}{2-cos^2(x)}$$.
I'm not sure if this right at all. My prowess with the quotient rule is shoddy at best.


text:
#### Archived

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# Importance sampling BRDFs

This topic is 5275 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi, Let''s say I have a BRDF function called F. We assume that over the hemisphere of directions, F always integrates to one (as in diffuse lighting), so no energy is lost in scaling through F. Now assume I have a BRDF function G = F*0.1 (that is, the tenth of F). G clearly doesn''t integrate up to one, but to 0.1. So 90% of the energy is lost when scaling through G! But when we importance sample (using rejection sampling, for example) either F or G, we get exactly the same distribution of samples. Therefore, using 1 as weight for every importance sampled direction, the importance sampled BRDF integrates to PI. We just divide it by PI, and tada, we always get a BRDF integrating to one! After this we can control the percentual amount of reflected light using a user-defined reflectance parameter. But when we use F or G not to generate samples, but simply to scale light, as is in computing direct lighting for example, we get totally different results, because G is after all only 10% of F. Of course we don''t have this kind of multiplications by 0.1 in practice, but simple changes of BRDF parameters (such as in Shclick BRDF) can cause the BRDF to not integrate to unity. So is there some quirk around this "BRDF doesn''t integrate to unity" problem? Probably based on weighting samples appropriately? - Mikko Kauppila

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text:
Extending Bayesian structural time-series estimates of causal impact to many-household conservation initiatives

# Extending Bayesian structural time-series estimates of causal impact to many-household conservation initiatives

## Abstract

Government agencies offer economic incentives to citizens for conservation actions, such as rebates for installing efficient appliances and compensation for modifications to homes. The intention of these conservation actions is frequently to reduce the consumption of a utility. Measuring the conservation impact of incentives is important for guiding policy, but doing so is technically difficult. However, the methods for estimating the impact of public outreach efforts have seen substantial developments in marketing to consumers in recent years as marketers seek to substantiate the value of their services. One such method uses Bayesian Stuctural Time Series (BSTS) to compare a market exposed to an advertising campaign with control markets identified through a matching procedure. This paper introduces an extension of the matching/BSTS method for impact estimation to make it applicable for general conservation program impact estimation when multi-household data is available. This is accomplished by household matching/BSTS steps to obtain conservation estimates and then aggregating the results using a meta-regression step to aggregate the findings. A case study examining the impact of rebates for household turf removal on water consumption in multiple Californian water districts is conducted to illustrate the work flow of this method.

\startlocaldefs\endlocaldefs\runtitle

Extending BSTS impact estimates to conservation programs

{aug}

, and

## 1 Introduction

Many government agencies offer economic incentives to homeowners to reduce the environmental impact of their households. Common incentives are subsides for utility reducing appliances or rebates for conservation actions, such as removing conventional lawns in drought regions. Agencies that create these programs are faced with two questions during reviews: (1) how effective is the marketing of the conservation program to households, and (2) by how much does the incentivized conservation technology or activity reduce consumption of the target utility. Answering the first question is valuable for assessing prior marketing strategies and improving future outreach efforts, while answering the second is key to assessing the value of extending the program.

Internet marketing analysts also perform impact assessments to estimate how effectively an advertising campaign increases the so-called conversion rate of the public, where online conversion rate is commonly quantified by clicks to an ad. google propose a method for this type of analysis, where a treatment market exposed to the new advertising campaign is matched to another, similar control market. Then, the relationship between these markets prior to the campaign is modeled using Bayesian Structural Time Series (BSTS), and a counterfactual estimate of clicks per ad is produced during the campaign for the treatment market. The difference between the treatment (actual) and control (counterfactual) gives the estimated impact of the campaign.

At the market level, internet marketers and conservation program managers are both interested in understanding if an outreach action was effective in stimulating engagement by individuals. Internet marketers are interested in clicks, while program managers are interested in participation in programs. Assuming that suitable benchmark markets can be found for conservation programs, it should be straightforward to apply the methodology introduced in google to assess the effectiveness of a conservation program’s outreach effort.

However, in this paper we will demonstrate how the matching and BSTS approach is particularly useful for estimating the impact of conservation programs on household consumption; in other words, for addressing the second question faced by agencies. To do so, the method proposed by google is applied to individual households. Instead of estimating the number of clicks generated by a marketing campaign, the method is used to estimate how many units of a utility are saved by a household’s participation in a conservation program. The method is further extended using a meta-regression step to systematically aggregate the estimates per household into general findings for informing policy.

The paper is developed as follows. First, in Section 2, a case study is introduced concerning rebate offers made to households for turf removal in California. This context will serve as the basis for summarizing the matching/BSTS methodology and its extension in Section 3. Then, in Section 4, the proposed impact estimation method is carried out and results are discussed. In Section 5, features of the method proposed in this paper and other approaches are discussed. Section 6 concludes.

## 2 California turf rebate data

With outdoor landscaping representing approximately half of urban water usage, the Californian water management community has identifed outdoor water usage in general mayer2015, and ornamental lawns specifically CUWCC2015 as a key opportunity in the larger effort to increase water conservation. Between July 2014 and April 2016, the Metropolitan Water District (MWD), the regional wholesaler of Colorado and Bay Delta water for Southern California, paid out $270.7 million directly for turf rebates under its regional program and another$15.1 million to supplement member agency spending on turf replacement. Metropolitan indirectly serves 6.1 million residential households across Southern California mwd2016. In addition, millions in local retailer turf rebate supplements have been paid out (for example in Los Angeles, Long Beach, San Diego and Moulton Niguel).

The data used in this paper to illustrate conservation program impact estimation was provided by 3 water utilities: Moulton Niguel Water District (MNWD), Irvine Ranch Water District (IRWD), and Eastern Municipal Water District (EMWD). Each utility provided two data sources. The first is a panel data set of monthly billed water usage and customer characteristics identified by account and service point (water meter) identifiers for single family households. The second is a data set detailing participation in water efficiency rebate programs, of which turf removals are the primary interest for this study.

These two data sets are merged, and any turf rebate instances tied to accounts that appear more than once are dropped to prevent over-counting. Observations with clearly extreme values for variables used in a modelling step were also identified and removed using visual inspection and simple multivariate robust regression models. The remaining accounts are then further restricted to those that have at least two years of data (24 observations) in the pre-rebate period and one year of data (12 observations) in the post-rebate period. The pre- and post-rebate periods are determined relative to the month that the post-rebate inspection was performed. Finally, the water districts make use of default values in cases where the actual value is unknown. Some districts substitute default values for irrigable area when actual values are not known. Customers with default values were dropped in cases where this was obvious due to bunching of many customers at the same value of irrigable area.

After cleaning, the California Turf Rebate Dataset (henceforth, CTRD) that serves as the working dataset in this paper contains 545 households that received either traditional or synthetic turf rebates. The variables are defined as follows:

• Customer ID: unique identifier for each household.

• Month and Year: the month and year of the water bill.

• HH Size: number of permanent residents at the property.

• Irr Area Sf and Rebate Quantity: the square feet of irrigable area and square feet of turf removed during rebate, respectively.

• Rebate Area Ratio: the proportion of turf area removed, calculated as .

• Evapotranspiration: The reference evapotranspiration, , in inches.

• Population density 2015: population density of the zip code where a household is located.

• Pre-turf removal efficiency: a measure of a household’s deviation from that household’s allocated water budget. A water budget is the quantity of water calculated by the district as adequate for a given household. With denoting the last time point prior to turf removal for treatment household, and that household’s consumption denoted as , the mean log ratio of used water over budgeted water at each period prior to turf removal is:

 1pp∑t=1logtrt+1[budget tr]t+1, (2.1)

where 1 is added to the usage and budget to avoid dividing by zero. Values below 1 indicate that the household is a water conserver prior to turf removal, while values above 1 indicate the opposite. This variable gives an indication of the water usage behavior of the household prior to turf removal.

## 3 Methodology

The matching/BSTS methodology outlined by google focuses on measuring the impact of a discrete marketing event. In the conservation program context, analogous example events are a rebate offer, a change in pricing structure or an advertising campaign; but the focus of this paper is on the non-analogous application of measuring the impact of a physical change to a household on utility usage. The aim of the matching/BSTS analysis is to quantify the event’s impact on a response metric of interest (e.g., household consumption of a utility). The causal impact of a conservation action, or treatment, is the difference between the observed value of the response and the (unobserved) value that would have been obtained under an alternative set of circumstances. Usually the alternative circumstances of interest are those that would have occurred had no marketing or monetary offering event occurred, or when no physical change to a household occurred. This case is frequently referred to as the counterfactual.

The innovation of the matching/BSTS methodology for causal impact estimation is the construction of a counterfactual based on two steps. First, one or more time series from a pool of candidates are classified by a matching algorithm as similar enough to the treated time series in the pre-treatment period that they can be used to infer trends that would influence the behavior in the treated series post-treatment. Second, using a BSTS, the relationship between the matched series and the treated series is modeled pre-treatment and used to predict the treated series post-treatment. This post-treatment prediction is the matching/BSTS counterfactual of the treated series under the scenario where no treatment was applied. The difference between the actual and predicted consumption of the treatment series is considered the impact of the treatment.

Many utilities collect data on utility consumption and conservation program participation at the consumer level. One can apply the methodology just described to a consumer that has participated in a conservation program. In this case, the methodology detailed in google and used in market impact analysis by online marketers is suitable. However, unlike a single market, a single consumer is rarely of much interest. Rather, it is desirable to apply this method to many users to gain deeper insights into broad consumer behaviors, and the behaviors of subgroups of consumers. One way to obtain an aggregated estimate of consumer behavior is to perform a meta-regression.

Meta-analysis is the practice of pooling estimates from multiple studies on the same effect to expose general properties of the effect. This approach is popular in the medical statistics literature, where it is common to pool estimates from studies of a medical phenomenon or treatments conducted in different regions or through different programs because there is typically reason to believe that a study in a single location will not completely explore the characterstics of the phenomenon or treatment due to cultural or institutional confounding. In the CTRD, estimates of turf removal impact on water consumption are obtained for 545 households that filed rebates. Each estimate of the rebate impact might be thought of as an estimate for the effect of a medical treatment obtained by a single study.

The steps below describe the matching/BSTS process and its extension in more detail in the context of the turf rebate case study data. Given 545 treatment accounts which participated in a turf removal rebate and are examined in this study:

1. Each treatment account which has participated in a turf rebate is matched with a set of control accounts from the same zip code which did not participate in a turf rebate. These control accounts are chosen by how similar their historical usage patterns are to the usage patterns of the treatment account , based on a weighted combination of their Pearson correlation and their warping distance.

2. After the have been chosen, we fit a BSTS model and use it to estimate the monthly impact of turf removal on water savings. The BSTS model uses the water usage patterns of the control accounts to create a synthetic control corresponding to the expected water usage of if there had been no turf removal. The predicted usage in the post-rebate period is then subtracted from observed usage to obtain a monthly water savings estimate for .

3. After water savings estimates have been calculated for each treatment account, the last step is to obtain an overall summary estimate. This is done with a meta-regression approach that uses the estimates and a measure of estimate accuracy from each treatment account as the inputs into a random effects model.

For the sake of brevity, this workflow will be referred to henceforth as M123; (1) Matching, (2) Modeling, (1) Meta. As shorthand, steps will be referred to as, for example, M1 for matching, or M13 for matching and meta. The first two steps are implemented into a workflow by the MarketMatching package. 1

### 3.1 Choosing Control Accounts

The first step in obtaining an estimate of the turf removal impact for account is to find accounts that did not remove their turf that show similar behavior to . Candidate accounts were identified by choosing controls from within the same zip code as . Within each zip code there may still be thousands of possible controls. These remaining possibilities are ranked by how similar their historical water usage patterns are to the historical usage of .

Account matching is often based on variables like property size, property value, or education levels. However, the importance of environmental attitudes, for example arising from public awareness actions and social change has been shown to influence water consumption hollis2016. The difficulty of incorporating these and other difficult-to-quantify factors driving household water usage, and the fairly stable water consumption patterns observed by most households, make matching based on water consumption patterns attractive. The premise of using historically predictive relationships between accounts to perform counterfactual analysis in this fashion has been advocated by, for example, abadie2010 and google.

Let and be a treatment and control time series with observations each for which a similarity ranking is desired. This similarity ranking is specified as a weighted composite of two other similarity measures. The first is the Pearson correlation:

 ρ(tr,c)=∑pt=1(trt−¯tr)(ct−¯c)√∑pt=1(trt−¯tr)2√∑pt=1(ct−¯c)2

The second ranks them according to their dynamic time warping (DTW) distance from . To compute the warping distance between two time series, we must identify the warping curve that has the minimum warping distance,

 D(tr,c)=p∑t=1d(ϕtr(t),ϕc(t))mϕ(t),

where is the local of the points at time after they have been remapped by the warping functions and , and is a per-step weight that control the slope of the warping curve. The calculation of the DTW distance is done using the dtw package in R. For details about the package and about dynamic time warping see giorgino2009.

Let the vector denote the similarity scores for candidate control accounts with respect to , where the th element of is given by:

 rk=(1−α)ρ(tri,ck)+αD(tri,ck);α∈[0,1]. (3.1)

Then, the control households corresponding to the first largest values in are used as controls for in the structural time series model for discussed in the next section.

### 3.2 Estimating Water Savings

A widely used approach for estimating the impact of interventions, like rebate offerings, is differences-in-differences. Taking this approach in the turf removal context, the estimated impact of turf removal on water savings is the difference between water usage when turf was removed, and the amount of water that would have been used if no turf had been removed bamezai1995.

To accurately estimate the reduction in water usage due to turf removal, a model for the counterfactual case needs to account for other variables determining water usage. Water use is determined by a multitude of factors, such as weather, household size, social perspectives on water usage, and turf removal. Covariates like weather and household size are measured by agencies and are straightforward to account for as covariates in a model.

This leaves the matter of accounting for dynamic behavioral patterns. Recognizing the need to address this aspect of water use, hollis2016 examined how variables measuring media factors, such as advertising volume, explain water use patterns. The inclusion of media presence explicitly in a usage model is desirable, but two issues that arise with this approach are properly quantifying media presence and accounting for the different levels of exposure experienced by water users.

Another way to account for dynamic behavior and classical covariates simultaneously is to explicitly model the counterfactual of a time series observed both before and after the rebate and use the resulting model to construct a synthetic control (cf. abadie2010). The approach of google is to construct a synthetic control by combining three sources of information using a state-space time-series model: (1) behavior of the pre-treatment target series (2) other time series that were predictive of the target series before the turf removal, and (3) in a Bayesian framework, prior knowledge about the model parameters, from earlier studies.

We will use static regression coefficients in our Bayesian structural time series model, which assumes that the linear usage relationship between the controls and the counterfactual expected usage for customers who did remove turf from their lawn remains fixed even after the turf is removed. Furthermore, we will allow for a local linear trend. For a treatment time series tr, this model has the form:

 trt=μtlevel+Ztregression+εt, Zt=β′x, μt+1=μt+δt+ημ,t% random walk and trend, δt+1=δt+ηδ,t% random walk for trend,

where , and . The regression component, captures the static linear relationship between the control series and the treatment series, while the level component captures local linear trends, enabling the model to react to unobserved sources of variability the control and treatment series are exposed to. Note that while the vector can contain covariates besides an intercept, classic covariates, such as household size are not explicitly included in the model implemented in this paper (nor in the model implemented by google), but are implicitly captured since the modelled trend is conditional on them.

By placing a spike-and-slab prior on the set of regression coefficients, and by allowing the model to average over the set of controls, it is possible to choose from many candidate controls george1997. To combine information about tr and the controls, the posterior distribution of the counterfactual time series is computed given the values of tr in the pre-intervention period, along with the values of the controls in the post-intervention period. Given a predicted and observed water use and , the difference yields a semi-parametric Bayesian posterior distribution for the water savings attributable to the turf removal at time , which can be used to obtain credible intervals. When water is considered saved, the value of is negative since less water was consumed than expected given the counterfactual water consumption . We take these estimates and adjust them from CCF/hundred cubic feet to gallons saved per square foot to obtain estimated monthly gallons saved per square foot of turf removed, calculated as:

 ˆgpsft=748.052×(trt−ˆtrt)% Rebate Quantity

The structural time series model was fitted using the CausalImpact package provided by Google for estimating the effectiveness of marketing campaigns google. A number of differences exist between the Google marketing context described in google, for which this approach was originally proposed, and the turf removal rebate context. Firstly, Google is able to assess the impact of the marketing campaign in terms of participation using this method, where participation is measured in number of clicks, because they have data on number of clicks prior to the campaign. It is in their interest to distinguish how many clicks after the start of the campaign were driven by the campaign, as opposed to organic. In contrast, prior to the rebate programs, the water districts did not track turf removal. The number of filed rebate claims before the start of the rebate programs is zero, so the rebates do not definitively show that turf removal has increased, unlike advertising clicks, which are monitored before and after an marketing action.

Another difference is that in the marketing context, the impact to estimate is the number of clicks generated as a consequence a marketing campaign, where a marketing campaign is either active or is not. The scale of the marketing campaign is not addressed. We could stop at estimating an average savings of a household that removes turf, but this neglects the important relationship between how much water use is reduced and the amount of the turf removed. To account for this, the estimated savings are divided by the square feet of turf removed, as calculated by utility staff in a post-rebate inspection (Equation 3.2). This allows for a normalized measure of rebate impact in terms of gallons per square foot of turf removed. Additionally, variables to quantify the magnitude of the turf removal are included in the meta-model in the final step.

An added complexity in this study is that in place of a single treatment cohort, or perhaps a few, hundreds of customers participated in the rebate program. The approach proposed in google stops at providing impact estimates on a single time series at a time. To obtain a broad overview of the impact of turf removal, it is desirable to aggregate estimates from all of the customers. This issue and the inclusion of the amount of turf removed in our framework will be addressed using a meta-regression approach. This meta-regression step will also allow us to explicitly extract the contribution of classical covariates, such as household size, whose influence is only implicitly modeled in the M2 step.

#### Example of M12

Figure 1 below shows two examples of the process described above. Specifically, the output of the matching process is shown through charts of water usage over time for the treatment household and its six closest matches. The output of the BSTS model is given by showing the actual and predicted consumption for the two examples. The example households were chosen for their wildly different behavior patterns in the post-rebate period. One household appears to cease outdoor watering completely after their turf removal, causing their usage to stabilize at winter levels and achieving an estimated 66% reduction in overall water use. The other example household shows a decrease in usage relative to its own past behavior, but shows no significant reduction compared to its similarly-behaving peers. This effect may be due to increased awareness of the California drought and the mandatory restrictions put in place in April 2015. Thus the water savings would be attributed to behavioral change among households in the region but not directly to the removal of turf.

#### Parameter selection

A number of parameters must be chosen when applying the matching procedure and BSTS model. We assessed these in terms of their impact on the mean water savings estimates obtained from the BSTS models.

A sensitivity analysis was performed to determine the effect of parameter choices at the matching stage on final estimates of water savings. Specifically, a random sample of 150 accounts were rerun under all combinations of the different parameter configurations visible in Table 1. While these are not the only parameters in the model, they are three of the ones most likely to impact the water savings estimates because they directly impact the choice of control accounts.

In the BSTS model, the value for in the local linear trend must also be selected. This is the local level standard deviation which controls the prior standard deviation of the local linear trend submodel. The local level term modifies how adaptable the model is to short term changes, and its standard deviation is important because it affects the breadth of the posterior intervals. google recommend that the value of 0.01 can be used when the relationship between the controls and the treatment is strong enough to obtain an informative model. The authors indicate that this is more likely when many control candidates are available. The water usage data set contains a large pool of control candidates, and matching results are typically strong (high values in the r vector, and visually convincing matches), so 0.01 is taken.

After calculating savings estimates under each parameter set, the mean of estimated savings for the sample under each parameter set was calculated. This gives an idea of how sensitive the matching process is to changes in the parameters. These estimates are visible below in Figure 2.

#### Computation

A draw-back of the M12 approach in conservation analyses is that the procedure needs to be applied to hundreds or thousands of households, but it is computationally heavy. This burden can be managed by cutting the time of individual operations or parallelizing them.

The most computationally expensive step in the M12 steps is the search for suitable matches. A major source of computational cost of this step is the DTW calculations used to obtain in Eq. 3.1. Figure 2 shows that there is some sensitivity in the meta-regression estimates (the analytical end-point of this methodology) as a function of , the DTW emphasis parameter. However, if the analyst reaches the conclusion that matching results are not sensitive to the emphasis on the DTW component, then a choice can be made to set in Equation 3.1 to 0. This eliminates the DTW calculations and substantially increases the computation speed of the procedure.

A second way to improve operation time is to parallelize the M12 steps. This topic is not addressed in google, possibly because the authors are addressing the normal situation in market impact analysis where an ad campaign is applied to one or a small number of markets and matches need to be found from perhaps a few hundred markets. Such a small number of M12 operations does not justify parallelization since even a single core can complete the task quickly. In contrast, when assessing the impact of a conservation action on hundreds or thousands of households, with thousands of candidates to match against, it becomes highly beneficial to take advantage of the distinctness of the household-level M12 calculations.

Two levels of the M12 steps can be parallelized. In the simpler approach, parallelization is performed at the participating household level according to the following steps:

Parallelization scheme 1

1. Select a participating household and candidates for matching (based on geographic proximity, zip code, etc).

2. Distribute the data for the selected participating and matching candidate households to a core.

3. perform the M12 steps and return the estimated conservation impact from the core for each time point in the post-rebate period.

4. Perform prior steps for all participating households as cores become available.

When the number of cores available is large and the number of candidates to match is extremely large, it may be advantageous to further distribute the computing as follows:

Parallelization scheme 2

1. Determine a number of cores that will be devoted to matching for a household .

2. Select a participating household

3. Select subsets of matching candidates (based on geographic proximity, zip code, etc), such that for any two subsets, , or nearly so, where only slight overlap occurs due to rounding.

4. Distribute the data for the selected participating and matching candidate households to core .

5. Perform the matching step and return the similarity index for each candiate in .

6. Combine the similarity indexes to obtain the vector and select the matching households.

7. Run the BSTS step and estimate the conservation impact series on an available core.

8. Perform prior steps for all participating households as cores become available.

Table 2 shows the values of the matching procedure parameters based on the results from the sensitivity analysis, as well as required minimum observation period lengths and matching pool sizes.

### 3.3 Meta-regression on the savings estimates

Monthly estimated water savings attributable to turf removal are obtained from each of the BSTS models, yielding a total of 10759 impact estimates for 545 households. Furthermore, a credible interval can be calculated for each of these estimates. In addition to considering this large collection of estimates in their own right, we can also model them to reveal general patterns linking estimated savings to known household characteristics. Doing so allows us to make a step from the household BSTS models, which do not distinguish between behavioral trends and the influence of covariates, to general insights into drivers of water savings that are classically of interest.

The aggregation of the household savings estimates can be accomplished with meta-regression, which allows us to include covariates that we expect to drive water consumption, such as household size, and to account for variability between the households. More specifically, we will use a mixed-effects model for , the savings of the th household at time ,:

 ˆgpsfi,t=β0+β1xi,1,t+⋯+βpxi,p,t+ui+εi,t,

where are coefficients for selected fixed effects, is a random intercept per household and is a random error term. One of the underlying assumptions of many modeling techniques is that each observation is measured with equal precision. However, in this context, we have an estimate of the precision for each water usage prediction, courtesy of the highest posterior density credible intervals returned by the BSTS. Each credible interval can be characterized by an estimated variance , which can be used to compute weights for each observation: . By applying these weights to the observations in the meta-regression, observations for which the BSTS model was able to provide good predictions are given the most influence. The meta-regression is performed using the metafor package metafor. The sources in metafor also provide background on the respective methods used in mata-analysis.

## 4 Case study: Analyzing the CTRD

Having outlined the steps for performing a conservation impact analysis using the M123 approach, we return to the CTRD. Our goal is to understand how turf removal and other variables influence household water consumption.

After matching and performing the BSTS step, we obtain an estimated change in water consumption per household that participated in the rebate program. This estimate is a quantification of the total difference between the expected and realized water consumption at each time period. To better understand how turf removal influences water usage, the meta-regression step can be used to decompose this change in consumption into factors of interest, such as the magnitude of turf removal and household size.

The meta-regression we conduct to aggregate the results from the Bayesian STS model estimates of the water savings from the th turf-removing household at time is a mixed effects model with the following fixed effects structure:

 μ gpsfi,t=αi+β0+β1×HH Sizei,t+β2×Pre-turf removal efficiencyi,t+β3×ln(Rebate Quantityi,t)+β4×ln(Rebate Area Ratioi,t)+β5×Household incomei,t+β6×Evapotranspirationi,t+β7×Population densityi,t+β8×sin(2π/12Monthi,t)+β9×cos(2π/12Monthi,t)+β10×sin(4π/12Monthi,t)+β11×cos(4π/12Monthi,t)+ui+εi,t,

where gpsf is the monthly savings in gallons per square foot. The trigonometric terms in the model account for seasonality and general time trends in savings not captured by the BSTS model in the pre-removal period. Month, in this model, is a unique number for each of the months in the study and runs from .

Table 3 contains the fixed effects estimates of the fitted model. Negative values indicate that as the value of the variable increases, water is increasingly saved. The response variable is estimated water savings from households that removed turf, and the intercept gives a baseline value of around 17 gallons saved per square foot per month. Household size is not significant, though the direction is positive. Pre-removal efficiency has a strong, negative effect. This result indicates that households which tend towards excessive consumption prior to turf-removal save more per square foot. The log of rebate quantity is positive and significant, indicating that there are diminishing returns per square foot removed. The variable rebate area ratio displays similar behavior.

Above the single household level, zip codes with higher median household incomes tend to save less water per square foot of turf removed. This may be because these households practice less conservative gardening to start with. Population is only significant at the 10% level, but the direction is that areas with denser population save less per square foot removed. Finally, higher ET values result in higher savings, which is because when ET is high, yards that would normally be watered heavily, are not watered at all.

The trigonometric effects used by the model to capture general time trends in water savings are significant. The fact that time terms are significant indicates that the BSTS model does not completely anticipate how monthly developments influence water usage in houses that removed turf relative to their controls. An explanation for this is that during the training period the BSTS models a relationship between control and treatment houses that have lawns which are generally fixed in size. After turf removal, the relationship between these houses changes; water consumption by houses that remove turf reacts differently in the dry months than before removal because less watering is required. This explanation also applies to the ET variable, which explains higher savings than the BSTS models expect because the water savings benefits of turf removal are not available when fitting the BSTS model since that is pre-removal.

In preliminary specifications of the model, dummy variables were included for the different districts in the data set. An attractive feature of a multi-district dataset is the potential that different implementation regimes will create conditions for a natural experiment. However, the district effects were insignificant. This is potentially due to the similarity of the districts in this study, which are geographically nearby in Southern California. A greater variety of districts in the dataset could offer the potential for more insights.

The first post-modelling analysis we conduct is a comparison of average household savings by year. We do this for the sample in this study by using the model to predict the household savings given their moderator variables. Predicted savings are then grouped by household and year and averaged. The resulting savings estimates give an impression of the distributions of savings outcomes that would be expected by an analyst or policy-maker on this population (Figure 3). We see that annual savings were about 20 gallons per square foot. However, by aggregating the monthly savings to an annual level, we lose important details about the savings patterns.

A more nuanced approach is to use the model to predict the monthly household savings. Overlaying the predictions are their quantiles, ranging from 5% to 95%. The savings pattern illustrated in Figure 4 is highly intuitive. The highest savings are in the months of July, August and September, reaching a monthly average reduction of 2.7 gallons per square foot. During the months of January, February, and March, the reduction is much smaller but still valuable at -1.5 gallons per square foot. Additionally, the plot depicts the skewness of the savings distribution. Turf removal nearly always results in some savings, and savings for most households that save more than the average fall within a bandwidth of one gallon more than the average, with a small proportion of extreme savers. Examining the predictions against their quantiles is similar to the practice advocated by Riley2011 of examining prediction intervals in random-effects meta-analysis in order to have a more complete overview of the range of potential treatment outcomes.

## 5 Discussion

The M123 results for the CTRD quantify the contribution of turf removal to water savings and elaborate on how these savings are realized. However, M123 is not the only approach for obtaining these estimates.

### 5.1 Time-Series vs. Traditional Matching

One remaining question of interest is whether time series matching on historical usage produces comparable results to traditional matching on static attributes. In order to address this question, the mean distance from each treatment account to its matched control accounts was compared to the mean distance from each treatment to its potential controls that were not matched.

Distance was calculated by standardizing the covariates for household size and irrigable area within each zip code and customer class. The mean Euclidean distance was then calculated between the treatment and each of the matched and unmatched groups. The results of this calculation are visible in Figure 5. One can see that matching on usage patterns tends to result, on average, in matches that are also similar in their household size and irrigable area. However, this was not universally true and manual inspection revealed a large variation even among the matched control accounts. This aligns with the intuition that static covariates do not capture all aspects of water usage, and that dissimilar accounts may have very similar water usage patterns.

### 5.2 BSTS vs. mixed model with dummies

The M123 approach uses a BSTS model to obtain . Consider instead an approach based on a linear mixed model with dummy variables for post-rebate observations

 usei,t=β0+β1x1,i,t+⋯+βpxp,i,t+δdi,t+ui+εi,t, (5.1)

where is a dummy variable indicating if a household has removed turf, and is a coefficient quantifying this effect. Given that we know the impact of turf rebates is conditional on climate factors, we could also expand the list of dummy variables to include interactions with season and ET. Along this line, many papers concerned with modeling water consumption use mixed models containing covariates thought to correlate with water usage and random intercepts to control for between-consumer variation (fielding2013, harlan2013, mitchell2013, olmstead2007, EndterStrata)

Although both M123 and the mixed model approach both include individual unit information in the model, they obtain estimates for different questions. The M123 estimates to what extent a participating household’s consumption is different from what it would have consumed if it had not participated. A mixed model estimates how consumption by participating households is different from non-participating households, by means of the dummy variable. The M12 approach is interesting if one is concerned with selection bias, wherein households that participate in conservation programs are likely to consume less water to begin with. The mixed model approach can not disentangle the a household’s concern for water use and water use reduction incurred by technical savings from a modification, such as turf removal. In contrast, the M12 approach makes some progress in this direction by modeling household water use idiosyncrasies in the pre-treatment phase that may be accounted for by unmeasurable preferences, such as concern for water use.

### 5.3 Improvement with quantile regression

Figure 4 gives a broader overview of the impact of turf removal than point estimates for a model of the mean, such as a table of coefficient estimates would. Ultimately, however, Figure 4 is a detailed view of the prediction intervals generated by the model of the mean shown in Table 3. It is of interest to understand better how the behavior of households in different water consumption brackets relate to descriptive covariates. Quantile regression provides one tool for exploring these relationships quantileRef2005. Employing a form of hierarchical spline models to estimate conditional quantiles, hendricks1992, explains that in studying electricity consumption in Chicago households, that the relationships between consumption and covariates differ across consumption quantiles. The methodology applied by the authors bears some resemblance to the M123 method in that models are fitted per household and the results are aggregated up to general results. As the authors state, this “hierarchical” linear models approach was introduced by lindley1972 and has been widely applied.

The M123 method differs from lindley1972, hendricks1992 and others in the methods used in the M12 steps. Regarding the M3 step, an option which post-dates hendricks1992 and is particularly suitable in concept is quantile regression with random effects quantileLongitudinal, fastQuantile2013. In application, functions from the rqpd package rqpd can be applied in the M3 step to perform the meta-regression. This approach was tested on the CTRD with some promising results, but the computational speed makes it prohibitively slow to apply to datasets of this size. Applying it to smaller datasets would be feasible, and there are prospects that methods such as fastQuantile2013 will allow for more rapid computation of these models in the near future.

## 6 Conclusions

The M123 methodology enables estimation of conservation program impacts using contextual customer attribute data of interest to policy makers and characterizations of usage dynamics based on observed household consumption. This is in contrast to approaches which rely purely on extensive lists of covariates that are at best proxies for consumption behavior.

In the CTRD case study the impact estimate is 24.6 gallons saved per year per square foot of turf removed. Bootstrapped standard errors of those predicted water savings are 0.11 gallons per year per square foot of turf removed. Those water savings are stable across district and vary sinusoidally over time highlighting the structural water savings of turf market transformation for regional and statewide water reliability initiatives. At $2 paid per square foot turf removed and assuming a hyperbolic discount rate of five percent over a landscape conversion lifespan of thirty years, that translates into a present value of$1422 plus or minus seven dollars per acre foot of water saved.

The M123 approach can be used to evaluate the water savings associated with other conservation rebates, other customer-level demand management interventions, and potentially other natural resource conservation programs in energy or natural gas. As the old adage goes, “you cannot manage what you cannot measure” and such rigorous impact evaluations can help public managers navigate the uncertainties of program design in conservation policy development.

Measuring savings at the household level, as in the CTRD, allows managers to highlight promising technical solutions, like turf removal, but also target educational materials on efficient consumption practices to customers that have seen dis-savings in their post-implementation period compared to their expected counterfactual ultility use. In the CTRD case, this would mean, for example, sending educational materials to households that removed turf, but have saved less water than expected: perhaps because they are not aware of proper watering practices for their new native landscapes.

## 7 Acknowledgements

This research was funded through the California Data Collaborative. The authors would like to thank the Moulton Niguel, Irvine Ranch, Eastern Municipal, Las Virgenes Municipal, Santa Margarita, and Monte Vista water districts, along with the Inland Empire Utilities Agency, the East Bay Municipal Utility District, and the Metropolitan Water District of Southern California for all of their support. The authors also thank Michael Hollis for his thoughtful comments and peer review.

## References

### Footnotes

1. The code was modified and is available at https://github.com/christophertull/MarketMatching/tree/usability-improvements
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## Engineering Mechanics: Statics & Dynamics (14th Edition)

$\omega_1=7.173 rad/s$
The required angular velocity can be determined as follows: First, we apply the equation of motion in normal direction $\Sigma F_n=ma_n$ [eq(1)] As $a_n=\omega_2^2 r=0.2\omega_2^2$ and $Wcos\theta=20(0.2)\omega_2^2$ $\implies cos\theta=\frac{0.2-0.03}{0.2}=0.85$ Now from eq(1), we obtain: $20(9.81)(0.85)=4\omega_2^2$ This simplifies to: $\omega_2=6.456 rad/s$ Now according to conservation of angular momentum $mv_1r_1+I\omega_1=mv_2r_2+I\omega_2$ [eq(2)] We know that $v=0.2\omega$ and $I=\frac{1}{2}mr^2=\frac{1}{2}(20)(0.2)^2=0.4Kg/m^2$ Now we plug in the known values in eq(2) to obtain: $20(0.2\omega_1)(0.17)+0.4\omega_1=20(0.2\omega_2)(0.2)+0.4\omega_2$ This simplifies to: $\omega_1=1.2\omega_2$ $\implies \omega_1=1.111\times 6.456=7.173 rad/s$


text:
# Capacitors in parallel or series

vcsharp2003
Homework Statement:
Every textbook has a very simple formula to determine equivalent capacitance when multiple capacitors are connected in parallel or series.

My question is: Does the formula apply to only parallel plate capacitors or capacitors of any shape/size?
Relevant Equations:
##\dfrac {1} {C_{eq}}= \dfrac {1} {C_1} + \dfrac {1} {C_2} + \dfrac {1} {C_3} + ...##, when capacitors are connected in series

##C_{eq}={C_1} + {C_2} + {C_3} + ...##, when capacitors are connected in parallel
To me it seems like the formula applies to capacitors of any shape or size, since textbooks never mention any limitations on capacitor type when stating these formulae.

## Answers and Replies

cmb
Any.

It is the capacitances that add, not the components/devices.

If you physically construct a load of capacitor devices together you might also find there is an inter-component and/or inter-connection capacitance that is created too, and you might need to add that in as well to make sense of the result (especially RF circuits).

DaveE, berkeman and vcsharp2003
vcsharp2003
Any.

It is the capacitances that add, not the components/devices.

If you physically construct a load of capacitor devices together you might also find there is an inter-component and/or inter-connection capacitance that is created too, and you might need to add that in as well to make sense of the result (especially RF circuits).

Can we connect a parallel plate capacitor with a spherical capacitor in series or parallel? Never seen such examples in Physics textbooks, so was just curious.

cmb
Of course.

But also (of course) you can't take one plate of a parallel plate and one sphere of a spherical capacitor and think there would be the sum of capacitances between them if held nearby each other, that would make up a different device. (I presume that might not needs saying, but just in case.)

Consider each capacitor to be a black box with two terminals. Makes no difference what is inside. But if you then open up the box then it wouldn't make the same capacitor if combined with the insides of another black box.

Some capacitors are better or worse at certain jobs, specifically they may become less capacitive and possibly even inductive as the frequency of an AC signal (or dV/dt of a transient) on them changes.

vcsharp2003
Homework Helper
Gold Member
@cmb are you talking about how closely positioned capacitors can interact via their fringing fields? yes ok I guess this type of interaction isn't negligible in extremely high frequencies and it depends on how each capacitor is internally designed.

cmb
I'm saying that physically 'real' capacitors will have inductance and a self resonance. There is no such thing as a 'perfect' pure capacitor, they have to have conductors to route the charge in and out, which adds inductance. Even a vacuum capacitor. Some capacitors, such as electrolytics, have significant inductance at very low frequency as well as big cyclic losses (low Q/high loss coefficient/high ESR) and are completely unsuitable for high frequency work, being used instead to stabilise DC voltages.

There is far far more to a capacitor than just its capacitance.

I mean, another effect, for example, is its power rating, or VAr (reactive power handling). If you push a capacitor beyond its VAr rating it may become substantially NOT a capacitor. So also depends on how it is used.

There are also voltage limits for use, not just arcing across the terminals and across the charged plates, but the dielectric in the capacitor itself will be non-linear to higher or lower electric fields (excludes vacuum capacitors, hence their use for high power RF).

vcsharp2003
But also (of course) you can't take one plate of a parallel plate and one sphere of a spherical capacitor and think there would be the sum of capacitances between them if held nearby each other, that would make up a different device. (I presume that might not needs saying, but just in case.)

So one could say that a spherical capacitor and a parallel plate capacitor cannot be connected in series or parallel so that the formula of equivalent capacitance applies. Is that correct?

Last edited:
Homework Helper
Gold Member
So one could say that a spherical capacitor and a parallel plate capacitor cannot be connected in series or parallel so that the formula of equivalent capacitance applies. Is that correct?
Back to why not? Each of the capacitors you describe, consists of at two conductors with two wire leads, one per conductor, coming out of the geometrical arrangement. Each capacitor by itself has its own capacitance which is strictly a geometric property. Call the capacitances ##C_1## and ##C_2##. You can connect one pair of the wires from one capacitor to the other pair any way you please.

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vcsharp2003
vcsharp2003
Back to why not? Each of the capacitors you describe, consists of at two conductors with two wire leads, one per conductor, coming out of the geometrical arrangement.

Very well explained. Thankyou.

kuruman
Gold Member
Homework Helper
Gold Member
OP wants us to consider two capacitors, one spherical and the other parallel-plate. The link you provided considers a single capacitor where one conductor is spherical and the other planar.

Homework Helper
2022 Award
What does the OP mean by a spherical capacitor?? The capacitance of an isolated sphere is not well localized. What do you mean @vcsharp2003?

vcsharp2003
What does the OP mean by a spherical capacitor?? The capacitance of an isolated sphere is not well localized. What do you mean @vcsharp2003?

A spherical capacitor consists of a hollow or a solid spherical conductor surrounded by another concentric hollow spherical conductor.

$$C= \dfrac {4\pi\epsilon R_1R_2} {R_2-R_1}$$

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hutchphd and Delta2
Homework Helper
2022 Award
OK then its explicitly localized and for usual low frequency work is just a capacitor as discussed.

Homework Helper
Gold Member
I am not an expert in capacitors but I think this spherical capacitor has zero fringing fields. Actually the only minimal fringing field would be because there must be a small hole in the outer shell through which the lead to the central conductor will pass.

Homework Helper
2022 Award
In fact I've never seen a spherical concentric "two plate" capacitor. And in practice the parallel plate ones are either rolled up into a cylinder or folded up in the sintering. I think for any two-wire component the effect fringe fields interfering is de minimis. Inductive cross talk is a bigger issue.
The standard textbook question is the capacity of an isolated sphere (relative to infinity) which is what would give a different answer hence my inquiry.

Homework Helper
Gold Member
Inductive cross talk is a bigger issue.
I don't understand what cross talk is this, how can there be cross talk if all the electric and magnetic field is inside the sphere. Capacitor can have inductance though at high frequencies since the magnetic field in the interior won't be negligible at such frequencies.

Homework Helper
2022 Award
I would guess that in practice the inductive coupling of the wiring will be the big issue for any capacitor conglomeration. You can't ground both sides. But the flux coupling for any of the designs would seem less than any self inductance.
Are there microwave cavities shaped like concentric spheres? Waveguides are a black art to me...

cmb
You can't ground both sides.
hmmm ... well you could via inductances, and any 'real' grounding connectors will have some inductance, leading to a resonant impedance.

But I think that is deviating from the OPs region of expectations and operating knowledge too far.

vcsharp2003
Gold Member
I'm seldom impressed by people that take an introductory physics problem and make it harder by saying "in the real world...". Your average introductory physics course actually isn't about the real world, it's about teaching concepts. Electrostatics doesn't really exist, there is no such thing as DC voltage, and, yes, everything has inductance and capacitance. But, what is the point of bringing all that up?

vcsharp2003
Homework Helper
2022 Award
I'm seldom impressed by people that
My particular inquiries were prompted by the OP mentioning a "spherical capacitor" because the concept of the capacitance of an isolated sphere was not well explained to my freshman self and I am unaware of any practical use of a spherical two wire capacitor so I was concerned about his question. I believe the OP received good answers, however. The question of fringing fields seemed superfluous to me, but no blood was shed...

cmb
I'm seldom impressed by people that take an introductory physics problem and make it harder by saying "in the real world...". Your average introductory physics course actually isn't about the real world, it's about teaching concepts. Electrostatics doesn't really exist, there is no such thing as DC voltage, and, yes, everything has inductance and capacitance. But, what is the point of bringing all that up?
There is no such thing as a pure capacitance. There is only 'the real world'. The OP was answered with the 'pure capacitance' theoretical physics answer directly, then they (to paraphrase) asked 'what if', which can only ever end up with a real world answer if they feel a theoretical answer is inadequate for them.

Of all electronic components whose characteristic behaviours are strongly bound to the physics of their particular technology, then capacitors are ones that exhibit such dependencies the most.