slides sales-forecasting-session1-web

Arthur CHARPENTIER - Sales forecasting.

Sales forecasting # 1

Arthur Charpentier

[email protected]

1


Agenda

Qualitative and quantitative methods, a very general introduction

• Series decomposition

• Short versus long term forecasting

• Regression techniques

Regression and econometric methods

• Box & Jenkins ARIMA time series method

• Forecasting with ARIMA series

Practical issues: forecasting with MSExcel

2


Somes references

Major reference for this short course, Pindyck, R.S. & Rubinfeld, D.L. (1997).Econometric models and economic forecasts. Mc Graw Hill.

“A forecast is a quantitative estimate about the likelihood of future events whichis developed on the basis of past and current information”.

3


Forecasting challenges ?

“With over 50 foreign cars already on sale here, the Japanese auto industry isn’tlikely to carve out a big slice of the U.S. market”. - Business Week, 1958

“I think there is a world market for maybe five computers”. - Thomas J. Watson,1943, Chairman of the Board of IBM

“640K ought to be enough for anybody”. - Bill Gates, 1981

“Stocks have reached what looks like a permanently high plateau”. - Irving Fisher,Professor of Economics, Yale University, October 16, 1929.

4


Challenge: use MSExcel (only) to build a forecast model

MSExcel is not a statistical software.

Specific softwares can be used, e.g. SAS, Gauss, RATS, EViews, SPlus, or morerecently, R (which is the free statistical software).

5


Macro versus micro ?

Macroeconomic Forecasting is related to the prediction of aggregate economicbehavior, e.g. GDP, Unemployment, Interest Rates, Exports, Imports,Government Spending, etc.

It is a very difficult exercice, which appears frequently in the media.

6


−4

−2

02

46

810

American Express

University ofNorth Carolina

Goldman SachsPNC Financial

Kudlow & co

Figure 1: Economic growth forecasts, from Wall Street Journal, Sept. 12, 2002,Q4 2002, Q1 2003 and Q2 2003.

7


Macro versus micro ?

Microeconomic Forecasting is related to the prediction of firm sales, industrysales, product sales, prices, costs...

Usually more accurate, and applicable to business manager...

Problem is that human behavior is not always rational: there is alwaysunpredictable uncertainty.

8


Short versus long term?

0 50 100 150 200 250

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0 50 100 150 200 250

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Figure 2: Forecasting a time series, with different models.

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160 180 200 220 240

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160 180 200 220 240

−4

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160 180 200 220 240

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Figure 3: Forecasting a time series, with different models.

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The Nasdaq index, 1971−2007

1970 1980 1990 2000

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0.00

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010

0020

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log re

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daq i

ndex

Figure 4: Forecasting financial time series.

11


Series decomposition

Decomposition assumes that the data consist of

data = pattern + error

Where the pattern is made of trend, cycle, and seasonality.

General representation isXt = f(St, Dt, Ct, εt)

where

• Xt denotes the time series value at time t,

• St denotes the seasonal component at time t, i.e. seasonal effect,

• Dt denotes the trend component at time t, i.e. secular trend,

• Ct denotes the cycle component at time t, i.e. cyclical variation,

• εt denotes the error component at time t, i.e. random fluctuations,

12



The secular trends are long-run trends that cause changes in an economic dataseries,

three different patterns can be distinguished,

• linear trend, Yt = α + βt

• constant rate of growth trend, Yt = Y0(1 + γ)t

• declining rate of growth trend, Yt = exp(α− β/t)

For the linear trend, adjustment can be obtained, introducing breaks for instance.

For constant rate of growth trend, note that in that caselog Yt = log Y0 + log(1 + γ) · t, which is a linear model on the logarithm of theserie.

13



For those two models, standard regression techniques can be used.

For declining rate of growth trend, log Yt = α− β/t, which is sometimes calledsemilog regression model.

The cyclical variations are major expansions and contractions in an economicseries that are usually greater than a year in duration

The seasonal effect cause variation during a year, that tend to be more or lessconsistent from year to year,

From an econometric point of view, a seasonal effect is obtained using dummyvariables. E.g for quaterly data,

Yt = α + βt + γ1∆1,t + γ2∆2,t + γ3∆3,t + γ4∆4,t

where ∆i,t is an indicator series, being equal to 1 when t is in the ith quarter,and 0 if not.

The random fluctuations cannot be predicted.

14


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Figure 5: Standard time series model, Xt.

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Figure 6: Standard time series model, the linear trend component.

16


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Figure 7: Removing the linear trend component Xt −Dt.

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Figure 8: Standard time series model, detecting the cycle on Xt −Dt.

18


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Figure 9: Standard time series model, Xt.

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Figure 10: Removing linear trend and seasonal component Xt −Dt − St.

20


Exogeneous versus endogenous variables

Model Xt = f(St, Dt, Ct, εt, Zt) can contain on exogeneous variables Z, so that

• St, the seasonal component at time t, can be predicted, i.e.ST+1, ST+2, · · · , ST+h

• Dt, the trend component at time t, can be predicted, i.e.DT+1, DT+2, · · · , DT+h

• Ct, the cycle component at time t, can be predicted, i.e.CT+1, CT+2, · · · , CT+h

• Zt, the exogeneous variables at time t, can be predicted, i.e.ZT+1, ZT+2, · · · , ZT+h

• but εt, the error component cannot be predicted

21


Exogeneous versus endogenous variables

Like in classical regression models: try to find a model Yi = Xiβ + εi which thehighest prediction value.

Classical ideas in econometrics: compare Yi and Yi, which should be as closed as

possible. E.g. minimizen∑

i=1

(Yi − Yi)2, which is the sum of squared errors, and

can be related to the R2, or MSE, or RMSE.

When dealing with time series, it is possible to add an endogeneous component.

Endogeneous variables are those that the model seeks to explain via the solutionof the system of equations.

The general model is thenXt = f(St, Dt, Ct, εt, Zt, Xt−1, Xt−2, ..., Zt−1, ..., εt−1, ...)

22


Comparing forecast models

In order to evaluate the accuracy - or reliability - of forecasting models, the R2

has been seen as a good measure in regression analysis,but the standard is theroot mean square error (RMSE), i.e.

RMSE =

√√√√ 1n

n∑i=1

(Yi − Yi)2

where is a good measure of the goodness of fit.

The smaller the value of the RMSE, the greater the accurary of a forecastingmodel.

23


●

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ESTIMATION

PERIOD

EX−POST

FORECAST

PERIOD

EX−ANTE

FORECAST

PERIOD

Figure 11: Estimation period, ex-ante and ex-post forecasting periods.

24


Regression model

Consider the following regression model, Yi = Xiβ + εi.

Call:

lm(formula = weight ~ groupCtl+ groupTrt - 1)

Residuals:

Min 1Q Median 3Q Max

-1.0710 -0.4938 0.0685 0.2462 1.3690

Coefficients:

Estimate Std. Error t value Pr(>|t|)

groupCtl 5.0320 0.2202 22.85 9.55e-15 ***

groupTrt 4.6610 0.2202 21.16 3.62e-14 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.6964 on 18 degrees of freedom

Multiple R-Squared: 0.9818, Adjusted R-squared: 0.9798

F-statistic: 485.1 on 2 and 18 DF, p-value: < 2.2e-16

25


Lest square estimation

Parameters are estimated using ordinary least squares techniques, i.e.β = (X ′X)−1X ′Y . E(β) = β.

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distan

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Figure 12: Least square regression, Y = a + bX.

26



Parameters are estimated using ordinary least squares techniques, i.e.β = (X ′X)−1X ′Y . E(β) = β.

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Figure 13: Least square regression, X = c + dY .

27



Assuming ε ∼ N (0, σ2), then V (β) = (X ′X)−1σ2.

The variance of residuals σ2 can be estimated using ε′ε/(n− k − 1).

It is possible to test H0 : βi = 0, then βi/σ√

(X ′X)−1i,i has a Student t

distribution under H0, with n− k − 1 degrees of freedom.

The p-value corresponding to the power of the t-test, i.e. 1- probability of secondtype error.

The confidence interval for βi can be obtained easilty as[βi − tn−k(1− α/2)σ

√[(X ′X)−1]i,i; βi + tn−k(1− α/2)σ

√[(X ′X)−1]i,i

]where tn−k(1− α/2) stands for the (1− α/2) quantile of the t distribution withn− k degrees of freedom.

28



3.5 4.0 4.5 5.0 5.5

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10.

010.

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030.

04

Endemics

Area ●

−0.15 −0.10 −0.05 0.00 0.05−0

.01

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0.02

0.03

0.04

Elevation

Area ●

29



The R2 is the correlation coefficient between series {Y1, · · · , Yn} and {Y1, · · · , Yn},where Yi = Xiβ. It can be interpreted as the ratio of the variance explained byregression, and total variance.

The adjusted R2, called R2, is defined as

R2

=(n− 1)R2 − k

n− k= 1− n− 1

n− k − 1(1−R2).

Assume that residuals are N (0, σ2), then Y ∼ N (Xβ, σ2I), and thus, it ispossible to use maximum likelihood technique,

logL(β, σ|X, Y ) = −n

2log(2π)− n

2log(σ2)− (Y −Xβ)′(Y −Xβ)

2σ2

Akake criteria (AIC) and Schwarz criteria (SBC) can be used to choose a model.

AIC = −2 logL+ 2k and SBC = −2 logL+ k log n

30



Fisher’s statistics can be used to test globally the significance of the regression,

i.e. H0 : β = 0, defined as F =n− k

k − 1R2

1−R2.

Additional tests can be run, e.g. to test normality of residuals, such asJarque-Berra statistics, defined as

BJ =n

6sk2 +

n

24[κ− 3]2,

where sk denotes the empirical skewness, and κ the empirical kurtosis. Underassumption H0 of normality, BJ ∼2 (2).

31


Residual in linear regression

1 2 3 4 5 6

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Fitted values

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lm(Y ~ X1 + X2)

Normal Q−Q

2

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1

32


Prediction in the linear model

Given a new observation x0, the predicted response is x′0β. Note that theassociated variance is V ar(x′0β) = x′0(X

′X)−1x0σ2.

Since the future observation should be x′0β+ε (where ε is unknown, but yieldadditional uncertainty), the confidence interval for this predicted value can becomputed as[βi − tn−k(1− α/2)σ

√1+x′0(X ′X)−1x0; βi + tn−k(1− α/2)σ

√1+x′0(X ′X)−1x0

]where again tn−k(1− α/2) stands for the (1− α/2) quantile of the t distributionwith n− k degrees of freedom.

Remark Recall that this is rather different compared with the confidenceinterval for the mean response, given x0, which is[

βi − tn−k(1− α/2)σ√

x′0(X ′X)−1x0; βi + tn−k(1− α/2)σ√

x′0(X ′X)−1x0

]

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Prediction in the linear model

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Regression, basics on statistical regression techniques

Remark statistical uncertainty and parameter uncertainty. Consider i.i.d.observations X1, lcdot, Xn from a N (µ, σ) distribution, where µ is unknown andshould be estimated.

Step 1: in case σ is known. The natural estimate of unkown µ is µ =1n

n∑i=1

Xi,

and the 95% confidence interval is[µ + u2.5%

σ√n

; µ + u97.5%σ√n

]where u2.5% = −1.9645 and u97.5% = 1.9645. Both are quantiles of the N (0, 1)distribution.

35


Regression, basics on statistical regression techniques

Step 2: in case σ is unknown. The natural estimate of unkown µ is still

µ =1n

n∑i=1

Xi, and the 95% confidence interval is

[µ + t2.5%

σ√n

; µ + t97.5%σ√n

]The following table gives values of t2.5% and t97.5% for different values of n.

36


n t2.5% t97.5% n t2.5% t97.5%

5 -2.570582 2.570582 30 -2.042272 2.042272

10 -2.228139 2.228139 40 -2.021075 2.021075

15 -2.131450 2.131450 50 -2.008559 2.008559

20 -2.085963 2.085963 100 -1.983972 1.983972

25 -2.059539 2.059539 200 -1.971896 1.971896

Table 1: Quantiles of the t distribution for different values of n.

This information is embodied in the form of a model - a single equationstructural model, a multiequation model, or a time series model

By extrapolating the models beyond the period over which they are estimated,we get forecasts about future events.

37


Regression model for time series

Consider the following regression model,

Yt = α + βXt + εt where εt ∼ N (0, σ2).

Step 1: in case α and β are known,

Given a known value XT+1, and if α and β are known, then

YT+1 = E(YT+1) = α + βXT+1

This yields a forecast error, εT+1 = YT+1 − YT+1. This error has two properties

• the forecast should be unbiased E(εT+1) = 0

• the forecast error variance is constant V (εT+1) = E(ε2T+1) = σ2.

38



Step 2: in case α and β are unknown,

The best forecast for YT+1 is then determined from a simple two-stage procedure,

• estimate parameters of the linear equation using ordinary least squares

• set YT+1 = α + βXT+1

Thus, the forecast error is then

εT+1 = YT+1 − YT+1 = (α− α) + (β − β)XT+1 − εT+1

Thus, there are two sources of error:

• the additive error term εT+1

• the random nature of statistical estimation

39


Figure 14: Forecasting techniques, problem of uncertainty related to parameterestimation.

40



Consider the following regression model

Goal of ordinay least squares, minimize∑N

I=1(Yi− Yi)2 where Y = α +βX. Then

β =n

∑XiYi −

∑Xi

∑Yi

n∑

X2i − (

∑Xi)

2

and

α =∑

Yi

n− β ·

∑Xi

n= Y − βX

The least square slope can be writen

β =∑

(Xi −X)(Yi − Y )∑(Xi −X)2

V (εT+1) = V (α) + 2XT+1cov(α, β) + X2T+1V (β) + σ2

41



under the assumption of the linear model, i.e.

• there exists a linear relationship between X and Y , Y = α + βX,

• the Xi’s are nonrandom variables,

• the errors have zero expected value, E(ε) = 0,

• the errors have constant variance, V (ε) = σ2,

• the errors are independent,

• the errors are normally distributed.

42


Regression model and Gauss-Markov theorem

Under the 5 first assumptions, the estimators α and β are the best (mostefficient) linear unbiased estimator of α and β, in the sense that they haveminimum variance, of all linear unbiased estimators (i.e. BLUE, best linearunbiased estimators).

The two estimators are further asymptotically normal,

√n(β − β)→N

(0,

n · σ2∑(Xi −X)2

)and

√n(α− α)→N

(0, σ2

∑X2

i∑(Xi −X)2

).

The asymptotic variances of α and β can be estimated as

V (β) =σ2∑

(Xi −X)2and V (α) =

σ2

n∑

(Xi −X)2

while the covariance is

cov(α, β) =−Xσ2∑

(Xi −X)2.

43


Regression model and Gauss-Markov theorem

Thus, if σ denotes the standard deviation of εT+1, the standard deviation s ofεT+1 can be estimated as

s2 = σ

(1+

1T

+(XT+1 −X)2∑

(Xi −X)2

)> σ.

44


RMSE (root mean square error) and Theil’s inequality

Recall that the root mean square error (RMSE), i.e.

RMSE =

√√√√ 1n

n∑i=1

(Yi − Yi)2

Another useful statistic is Theil inequality coefficient defined as

U =

√√√√ 1T

n∑i=1

(Yi − Yi)2√√√√ 1T

n∑i=1

Y 2i +

√√√√ 1T

n∑i=1

Y 2i

From this normalization U always fall between 0 and 1. U = 0 is a perfect fit,while U = 1 means that the predictive performance is as bad as it could possiblybe.

45


Step 3, assume that α, β and XT+1 are unknown, but thatXT+1 = XT+1 + uT+1, where uT+1 ∼ N (0, σ2

u). The two errors are uncorrelated.

Here, the error of forecast is

εT+1 = YT+1 − YT+1 = (α− α) + (β − β)XT+1 − εT+1

It can be proved (easily) that E(εT+1) = 0. But its variance is slightly morecomplecated to derive

V (εT+1) = V (α) + 2XT+1cov(α, β) + (X2T+1+σ2

u)V (β) + σ2+β2σ2u

And therefore, the forecast error variance is then

s2 = σ

(1 +

1T

+(XT+1 −X)2 + σ2

u∑(Xi −X)2

+ β2σ2u

)> σ2,

which,again, increases the forecast error.

46


To go further, multiple regression model

In the multiple regression model, Y = Xβ + ε, in which

Y =

Y1

Y2

...

Yn

,X =

X1,1 X2,1 ... Xk,1

X1,2 X2,2 ... Xk,2

... ... ...

X1,n X2,n ... Xk,n

,β =

β1

β2

...

βK

,ε =

ε1

ε2

...

εn

• there exists a textcolorbluelinear relationship between X1, , Xk and Y ,

Y = α + β1X1 + +βkXk,

• the Xi’s are nonrandom variables, and moreover, there are no exact linearrelationship between two and more independent variables,

• the errors have zero expected value, E(ε = 0,

• the errors have constant variance, var(ε) = σ2,

• the errors are independent,

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• the errors are normally distributed.

The new assumption here is that “there are no exact linear relationship betweentwo and more independent variables”.

If such a relationship exists, variables are perfectly collinear, i.e. perfectcollinearity.

From a statistical point of view, multicollinearity occures when two variables areclosely related. This might occur e.g. between two series {X2, X3, · · · , XT } and{X1, X2, · · · , XT−1} with strong autocorrelation.

48


To go further, forecasting with serial correlated errors

In previous model, errors were homoscedastic. A more general model is obtainedwhen errors are heteroscedastic, i.e. non-constant variance. Goldfeld-Quandt testcan be performed.

An alternative is to assume serial correlation. Cochrane-Orcutt or Hildreth-Luprocedures can be performed.

Consider the following regression model,

Yt = α + βXt + εt where εt = ρεt − 1 + ηt

with −1 ≤ ρ ≤ +1 and ηt ∼ N (0, σ2).

Step 1, assume that α, β and ρ are known.

YT+1 = α + βXT+1 + εT+1 = α + βXT+1 + ρεT

assuming that εT+1 = ρεT . Recursively,

εT+2 = ρεT+1 = ρ2εT

49


εT+3 = ρεT+2 = ρ3εT

εT+h = ρεT+h−1 = ρhεT

Since |ρ| < 1, ρh approaches 0 as h gets arbitrary large. Hence, the informationprovided by serial correlation becomes less and less usefull.

YT+1 = α(1− ρ) + βXT+1 + ρ(YT − βXT )

Since YT = α + βXT + εT , then

YT+1 = α + βXT+1 + ρεT

Thus, the forecast error is then

εT = YT − YT = ρεT − εT+1

50


To go further, using lag models

We have mentioned earlier that when dealing with time series, it was possible notonly to consider the linear regression of Yt on Xt, but to consider lagged variates

• either Xt−1, Xt−2, Xt−2, ...etc,

• or Yt−1, Yt−2, Yt−2, ...etc,

First, we will focuse on adding lagged explanatory exogneous variable, i.e.models such as

Yt = α + β0Xt + β1Xt−1 + β2Xt−2 + · · ·+ βhXt−h + · · ·+ εt.

Remark In a very general setting Xt can be a random vector in Rk.

51


To go further, a geometric lag model

Assume that weights of the lagged explanatory variables are all positive anddecline geometrically with time,

Yt = α + β(Xt + ωXt−1 + ω2Xt−2 + ω3Xt−3 + · · ·+ ωhXt−h + · · ·

)+ εt,

with 0 < ω < 1.

Note that

Yt−1 = α+β(Xt−1 + ωXt−2 + ω2Xt−3 + ω3Xt−4 + · · ·+ ωhXt−h−1 + · · ·

)+εt−1,

so thatYt − ωYt−1 = α(1− ω) + βXt + ηt

where ηt = εt − ωεt−1.

Rewriting Yt = α(1− ω) + ωYt−1 + βXt + ηt.

52


To go further, a geometric lag model

This would be called single-equation autoregressive model, with a single laggeddependent variable.

The presence of a lagged dependent variable in the model causes ordinaryleast-squares parameter estimates to be biased, although they remain consistent.

53


Estimation of parameters

In classical linear econometrics, Y = Xβ + ε, with ε ∼ N (0, σ2). Thenβ = (X ′X)−1X ′Y

• is the ordinary least squares estimator, OLS,

• is the maximum likelihood estimator, ML.

Maximum likelihood estimator is consistent, asymptotically efficient, and(asymptotic) variances can be determined. This can be obtined usingoptimization techniques.

Remark it is possible to use generalized method of moments, GMM.

54


To go further, modeling a qualitative variable

In some case, the variable of interest is not necessarily of price (continuousvariable on R), but a binary variable.

Consider the following regression model Yi = α + βXi + εi, with Yi =

1

0where the ε are independent random variables, with 0 mean.

Then E(Yi) = α + βXi.

Note that Yi is then a Bernoulli (binomial) distribution.

Classical models are either the probit or the logit model.

The idea is that there exists a continuous latent unobservable Y ∗ such that

Yi =

1 if Y∗i > ti

0 if Y∗i ≤ ti

with Y ∗i = α + βXi + εi, which is now a classical

regression model.

Equivalently, it means that Yi is then a Bernoulli (binomial) distribution B(pi)

55


wherepi = F (α + βXi),

where F is a cumulative distribution function. If F is the cumulative distributionfunction of N(0,1), i.e.

F (x) =1√2π

∫ x

−∞exp

(−z2

2

)dz,

which is the probit model, or the cumularive distribution of the logisticdistribution

F (x) =1

1 + exp(−x)

for the logit model.

Those models can be extended to so-called ordered probit model, where Y candenote e.g. a rating (AAA,BB+, B-,...etc).

Maximum likelihood techniques can be used.

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Modeling the random component

The unpredictible random component is the key element when forecasting. Mostof the uncertainty comes from this random component εt.

The lower the variance, the smaller the uncertainty on forecasts.

The general theoritical framework related to randomness of time series is relatedto weakly stationary.

57

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