{ From user Lonnie, Model Statistical_function at Thu, Aug 16, 2007 11:38 AM~~
}
Softwareversion 4.0.0
{ System Variables with non-default values: }
Samplesize := 200
Sampletype := 2
Typechecking := 1
Checking := 1
Saveoptions := 2
Savevalues := 0
Sampleweighting := adj_wt
Windstate Sampleweighting: 2,102,90,481,381
{!40000|Att_contlinestyle Graph_primary_valdim: 1}
Attribute Reference
Attribute Date_bough
Askattribute Recursive,Function,Yes
Model Statistical_function
Title: Statistical Functions in Analytica 4.0
Description: Analytica User Group session #3~
Statistical Functions in Analytica 4.0~
Aug 16, 2007~
~
A statistical function is a function that process a data set containi~~
ng many sample points, computing a "statistic" that summarizes the da~~
ta. Simple examples are Mean and Variance, but more complex examples ~~
may return matrices or tables. In this talk, I'll review statistical ~~
functions that are built into Analytica 4.0. In Analytica 4.0, all bu~~
ilt-in statistical functions can now be applied to historical data se~~
ts over an arbitrary index, as well as to uncertain samples (the Run ~~
index), eliminating the need for separate function libraries. I will ~~
demonstrate this use, as well as several new statistical functions, e~~
.g., Pdf, Cdf, Covariance. I will explain how the domain attribute sh~~
ould be utilized to indicate that numeric-valued data is discrete (su~~
ch as integer counts, for example), and how various statistical funct~~
ions (e.g., Frequency, GetFract, Pdf, Cdf, etc) make use of this info~~
rmation. In the process, I'll demonstrate numerous examples using the~~
se functions, such things as inferring sample covariance or correlati~~
on matricies from data, quickly histogramming arbitrary data and usin~~
g the coordinate index setting to plot it, or using a weighted Freque~~
ncy for rapid aggregation. ~
~
In addition, all statistical functions in Analytica 4.0 can compute w~~
eighted statistics, where each point is assigned a different weight. ~~
I'll cover the basics of sample weighting, and demonstrate some simpl~~
e examples of using this for computing a Bayesian posterior and for i~~
mportance sampling from an extreme distribution. ~
Author: Lonnie
Date: Thu, Aug 16, 2007 8:42 AM
Saveauthor: Lonnie
Savedate: Thu, Aug 16, 2007 11:38 AM
Defaultsize: 48,24
Diagstate: 1,1,0,550,300,17
Windstate: 2,20,36,620,415
Fontstyle: Arial, 15
Fileinfo: 0,Model Statistical_function,2,2,0,0,W:\TestModels\User Grou~~
p Webinar - Statistical Functions.ANA
Module Historical_data
Title: Historical Data
Author: Lonnie
Date: Thu, Aug 16, 2007 9:55 AM
Defaultsize: 48,24
Nodelocation: 72,48,1
Nodesize: 48,24
Diagstate: 1,1,0,504,395,17
Index Trading_date
Title: Trading Date
Definition: [37.847K,37.846K,37.845K,37.842K,37.841K,37.84K,37.839K,37~~
.838K,37.835K,37.834K,37.833K,37.832K,37.831K,37.828K,37.827K,37.826K~~
,37.825K,37.824K,37.821K,37.82K,37.819K,37.818K,37.817K,37.814K,37.81~~
3K,37.812K,37.811K,37.81K,37.807K,37.806K,37.804K,37.803K,37.8K,37.79~~
9K,37.798K,37.797K,37.796K,37.793K,37.792K,37.791K,37.79K,37.789K,37.~~
786K,37.785K,37.784K,37.783K,37.782K,37.779K,37.778K,37.777K,37.776K,~~
37.775K,37.772K,37.771K,37.77K,37.769K,37.765K,37.764K,37.763K,37.762~~
K,37.761K,37.758K,37.757K,37.756K,37.755K,37.754K,37.751K,37.75K,37.7~~
49K,37.748K,37.747K,37.744K,37.743K,37.742K,37.741K,37.74K,37.737K,37~~
.736K,37.735K,37.734K,37.733K,37.73K,37.729K,37.728K,37.727K,37.726K,~~
37.723K,37.722K,37.721K,37.72K,37.719K,37.715K,37.714K,37.713K,37.712~~
K,37.709K,37.708K,37.707K,37.706K,37.705K,37.702K,37.701K,37.7K,37.69~~
9K,37.698K,37.695K,37.694K,37.693K,37.692K,37.691K,37.688K,37.687K,37~~
.686K,37.685K,37.684K,37.681K,37.68K,37.679K,37.678K,37.677K,37.674K,~~
37.673K,37.672K,37.671K,37.667K,37.666K,37.665K,37.664K,37.663K,37.66~~
K,37.659K,37.658K,37.657K,37.656K,37.653K,37.652K,37.651K,37.65K,37.6~~
49K,37.646K,37.645K,37.644K,37.643K,37.642K,37.639K,37.638K,37.637K,3~~
7.636K,37.632K,37.631K,37.63K,37.629K,37.628K,37.625K,37.624K,37.623K~~
]
Nodelocation: 96,48,1
Nodesize: 48,24
Numberformat: 2,DD,2,2,0,0,2,0,$,0,"ABBREV",0
{!40000|Att_graphindexrange: 1,,0,,,0}
{!40000|Att_previndexvalue: [37.847K,37.846K,37.845K,37.842K,37.841K,37.84K~~
,37.839K,37.838K,37.835K,37.834K,37.833K,37.832K,37.831K,37.828K,37.827K~~
,37.826K,37.825K,37.824K,37.821K,37.82K,37.819K,37.818K,37.817K,37.814K~~
,37.813K,37.812K,37.811K,37.81K,37.807K,37.806K,37.804K,37.803K,37.8K~~
,37.799K,37.798K,37.797K,37.796K,37.793K,37.792K,37.791K,37.79K,37.789K~~
,37.786K,37.785K,37.784K,37.783K,37.782K,37.779K,37.778K,37.777K,37.776K~~
,37.775K,37.772K,37.771K,37.77K,37.769K,37.765K,37.764K,37.763K,37.762K~~
,37.761K,37.758K,37.757K,37.756K,37.755K,37.754K,37.751K,37.75K,37.749K~~
,37.748K,37.747K,37.744K,37.743K,37.742K,37.741K,37.74K,37.737K,37.736K~~
,37.735K,37.734K,37.733K,37.73K,37.729K,37.728K,37.727K,37.726K,37.723K~~
,37.722K,37.721K,37.72K,37.719K,37.715K,37.714K,37.713K,37.712K,37.709K~~
,37.708K,37.707K,37.706K,37.705K,37.702K,37.701K,37.7K,37.699K,37.698K~~
,37.695K,37.694K,37.693K,37.692K,37.691K,37.688K,37.687K,37.686K,37.685K~~
,37.684K,37.681K,37.68K,37.679K,37.678K,37.677K,37.674K,37.673K,37.672K~~
,37.671K,37.667K,37.666K,37.665K,37.664K,37.663K,37.66K,37.659K,37.658K~~
,37.657K,37.656K,37.653K,37.652K,37.651K,37.65K,37.649K,37.646K,37.645K~~
,37.644K,37.643K,37.642K,37.639K,37.638K,37.637K,37.636K,37.632K,37.631K~~
,37.63K,37.629K,37.628K,37.625K,37.624K,37.623K]}
Variable Stock_price
Title: Stock Price
Definition: Table(Trading_date,Stock)(~
119.9,28.1,4.65,19.18,23.22,~
124.03,28.27,4.7,19.35,23.8,~
127.79,28.63,4.73,19.72,24.02,~
125,28.71,4.68,19.99,23.98,~
126.39,29.3,4.74,20.09,23.92,~
134.01,30,5,20.2,24.68,~
135.03,29.55,5,19.68,24.13,~
135.25,29.54,4.96,20.08,24.13,~
131.85,28.96,4.91,19.66,23.91,~
136.49,29.52,5.14,20.09,24.3,~
135,29.3,5.1,19.79,23.8,~
131.76,28.99,5.1,19.12,23.62,~
141.43,29.4,4.89,19.58,23.85,~
143.85,29.39,4.92,19.62,23.54,~
146,29.98,4.98,20.01,24,~
137.26,30.71,5.24,20.58,24.5,~
134.89,30.8,5.23,20.64,24.53,~
143.7,31.19,5.29,20.78,24.72,~
143.75,31.16,5.33,20.61,24.55,~
140,31.51,5.4,20.6,25.26,~
138.12,30.92,5.28,20.41,25.06,~
138.91,30.78,5.29,20.38,26.33,~
138.1,30.03,5.34,20.2,25.95,~
137.73,29.82,5.37,20.4,25.97,~
134.07,30.07,5.43,20.5,26,~
132.39,29.49,5.38,19.98,24.57,~
132.35,29.33,5.32,19.72,24.97,~
130.33,29.87,5.33,20.16,24.96,~
132.3,29.97,5.38,20.4,24.68,~
132.75,29.99,5.33,20.49,24.6,~
127.17,30.02,5.22,20.07,24.59,~
121.26,29.74,5.19,19.92,24.27,~
122.04,29.47,5.26,19.71,23.74,~
120.56,29.83,5.16,19.85,23.92,~
121.89,29.87,5.07,19.69,23.79,~
119.65,29.52,5.01,19.16,23.38,~
122.34,29.49,5.02,19.48,23.48,~
123,29.49,5.08,19.39,23.7,~
123.9,30.22,5.13,19.68,24.29,~
121.55,30.01,5.05,19.53,23.94,~
123.66,30.46,5.1,19.88,24.1,~
125.09,30.51,5.05,19.79,24.17,~
120.5,30.49,5.05,19.86,24.24,~
118.75,30.52,5.08,19.64,23.23,~
117.5,30.39,4.99,19.3,22.67,~
120.38,29.85,4.92,18.84,22.2,~
120.19,30.02,4.96,19.21,21.93,~
124.49,30.05,5.04,19.06,21.83,~
124.07,29.62,4.97,18.73,21.31,~
123.64,30.29,5.02,19.35,21.49,~
122.67,30.58,5.07,19.48,21.96,~
121.33,30.72,5.11,19.67,22.16,~
118.4,30.59,5.18,19.66,22.36,~
121.19,30.69,5.1,19.38,22.18,~
118.77,31.11,5.12,19.42,22.08,~
114.35,30.79,5.06,19.31,22.3,~
113.62,30.48,5.16,19.24,22.16,~
110.69,30.17,5.15,18.75,21.97,~
112.89,30.58,5.38,19.16,22.67,~
113.54,30.69,5.38,19.37,22.99,~
111.98,31.05,5.39,19.32,22.63,~
110.02,30.83,5.29,19.25,22.7,~
109.44,30.98,5.3,19.05,22.23,~
107.34,31.07,5.12,18.99,22.18,~
107.52,30.9,5.14,18.84,22.01,~
109.36,30.97,5.22,18.94,22.12,~
108.74,30.89,5.15,18.98,22.28,~
107.34,30.58,5.13,18.49,22.21,~
106.88,30.78,5.2,18.83,22.47,~
105.06,30.75,5.22,18.95,22.15,~
103.92,30.71,5.22,19.05,21.96,~
100.81,30.56,5.24,19.03,21.9,~
100.4,30.97,5.18,19.02,21.74,~
100.39,30.61,5.15,18.86,21.93,~
99.47,30.4,5.09,18.59,21.8,~
99.8,29.94,5.22,18.8,21.5,~
99.92,30.12,5.26,19.1,21.87,~
98.84,29.1,5.25,18.95,22.09,~
95.34999999999999,28.99,5.27,18.9,22.26,~
93.24,28.79,5.94,18.82,21.94,~
93.51000000000001,28.78,5.92,18.94,21.91,~
90.97,29.02,5.93,19,22.16,~
90.27,28.69,5.89,18.76,21.81,~
90.40000000000001,28.6,5.94,18.73,21.35,~
90.34999999999999,28.85,5.95,18.89,20.98,~
91.43000000000001,28.73,5.86,18.9,20.69,~
90.24,28.61,5.8,18.63,20.46,~
92.19,28.54,5.9,18.7,20.5,~
92.59,28.11,5.9,18.59,20.47,~
94.25,28.4,5.93,18.85,20.68,~
93.65000000000001,28.57,5.93,18.57,20.1,~
94.68000000000001,28.55,5.93,18.67,19.58,~
94.27,28.5,5.86,18.56,19.38,~
94.5,27.87,5.79,18.36,19.31,~
93.65000000000001,27.74,5.8,18.14,19.13,~
92.91,27.87,6.01,18.13,19.13,~
93.75,27.75,5.96,18.16,19.09,~
93.24,27.64,6.05,18.17,18.86,~
95.45999999999999,27.72,6.06,18.49,19.06,~
95.84999999999999,28.22,6.22,18.39,19.29,~
93.52,28.02,6.19,18.24,19.27,~
93.95999999999999,28.27,6.31,18.49,19.16,~
93.87000000000001,28.52,6.49,18.17,19.34,~
91.48,27.84,6.2,17.55,18.99,~
91.13,27.83,6.22,17.18,19.11,~
89.59,27.33,6.24,16.7,19.15,~
89.56999999999999,27.28,6.22,16.72,19.14,~
90,27.4,6.31,16.88,19.23,~
88.40000000000001,26.72,6.15,16.65,19.12,~
89.87000000000001,27.44,6.31,17.07,19.48,~
87.97,27.29,6.27,16.63,19.1,~
88,27.32,6.21,16.69,19.23,~
87.72,27.61,6.17,16.49,19.12,~
88.19,27.83,6.25,16.88,19.4,~
86.31999999999999,27.55,6.07,16.37,19.11,~
85.41,27.76,6.06,16.71,19.22,~
87.06,28.09,6.04,16.77,19.59,~
84.61,28.17,6.13,16.43,19.86,~
83.93000000000001,27.87,5.98,16.29,20.03,~
88.51000000000001,29.07,6.27,16.82,20.85,~
89.06999999999999,28.9,6.27,16.82,20.76,~
89.51000000000001,29.39,6.27,17.27,20.97,~
89.2,29.35,6.28,17.2,20.88,~
85.90000000000001,28.83,6.38,16.98,21.18,~
84.83,28.74,6.29,16.7,21.23,~
85.20999999999999,29.46,6.31,16.91,21.31,~
85.3,29.4,6.4,16.77,21.14,~
84.7,29.01,6.4,16.62,20.87,~
84.88,28.94,6.42,16.65,20.8,~
83.27,28.98,6.48,16.7,21.03,~
86.18000000000001,29.26,6.59,16.71,21.36,~
86.15000000000001,29.37,6.59,16.91,21.51,~
84.15000000000001,29.51,6.53,17.02,21.31,~
83.94,29.61,6.64,17.16,21.28,~
84.75,30.19,6.63,17.42,21.23,~
84.74,30.56,6.55,17.05,21.11,~
85.73,30.86,6.64,17.16,20.96,~
85.55,30.48,6.43,17.16,20.93,~
85.94,30.53,6.35,17.27,20.89,~
85.38,30.6,6.33,17.15,20.53,~
86.25,30.45,6.34,16.98,20.6,~
86.7,31.09,6.15,17.14,20.84,~
85.7,30.74,5.66,17.12,20.55,~
86.79000000000001,30.72,5.75,17,20.79,~
88.5,31.11,5.77,17.27,20.82,~
89.06999999999999,31,5.79,17.12,20.65,~
94.95,31.1,5.99,17.52,21.04,~
97.09999999999999,31.16,6.06,17.3,22.3,~
94.62000000000001,31.21,6.06,17.5,22.13,~
95.8,30.7,6.14,17.39,21.92,~
97,29.66,6,17.77,21.52,~
92.56999999999999,29.96,5.73,17.82,21.03,~
85.47,29.93,5.65,17.86,21.01,~
85.05,29.64,5.6,17.64,21.1,~
85.66,29.81,5.69,17.68,21.17,~
83.8,29.86,5.54,17.51,20.35~
)
Nodelocation: 208,48,1
Nodesize: 48,24
Defnstate: 2,574,8,453,729,0,MIDM
Valuestate: 2,413,22,552,575,1,MIDM
Graphsetup: {!40000|Att_contlinestyle Graph_primary_valdim:4}~
{!40000|Graph_symbolsizekey:0}~
{!40000|Att_graphvaluerange Stock_price:1,,,,0}
Reformdef: [Stock,Trading_date]
Reformval: [Trading_date,Undefined]
{!40000|Att_xrole: -5}
{!40000|Att_yrole: -3}
{!40000|Att_coordinateindex: Stock}
Index Stock
Title: Stock
Definition: ['AAPL','MSFT','SUNW','ORCL','INTC']
Nodelocation: 96,112,1
Nodesize: 48,24
{!40000|Att_previndexvalue: ['AAPL','MSFT','SUNW','ORCL','INTC']}
Variable Mean_sp
Title: mean sp
Definition: Mean( Stock_price, Trading_date )
Nodelocation: 208,112,1
Nodesize: 48,24
Valuestate: 2,409,36,416,303,0,MIDM
Variable Correlation_sp
Title: correlation sp
Definition: Correlation( Stock_price, Stock_price[Stock=Stock2], Tradi~~
ng_date )
Nodelocation: 208,184,1
Nodesize: 48,24
Valuestate: 2,419,79,416,303,0,MIDM
Reformval: [Stock,Stock2]
Index Stock2
Title: Stock2
Definition: CopyIndex(Stock)
Nodelocation: 96,184,1
Nodesize: 48,24
{!40000|Att_previndexvalue: ['AAPL','MSFT','SUNW','ORCL','INTC']}
Variable Cov_mat
Title: cov mat
Definition: Covariance( Stock_price, Stock_price[Stock=Stock2], Tradin~~
g_date )
Nodelocation: 208,240,1
Nodesize: 48,24
Valuestate: 2,499,105,416,303,0,MIDM
Reformval: [Stock,Stock2]
Library Multivariate_distrib
Title: Multivariate Distributions
Description: A library of multivariate distributions.~
~
In a multivariate distribution, each sample is a vector. This vector~~
is identified by an index, identified by the I parameter of the func~~
tions in this library. A Mid value from a distribution function will~~
therefore be indexed by I, whlie a Sample from a distribution functi~~
on is indexed by both I and Run. These distribution functions can al~~
so be used from within the Random function to generate a single monte~~
-carlo sample, which will be indexed by I.~
~
This library also contains functions for generating correlated distri~~
butions. Correlate_with, for example, allows you to generate a univa~~
rite distribution with an arbitrary marginal distribution that has a ~~
specified rank correlation with an arbitrary reference distribution. ~~
Several functions may be used for generating serial correlations, w~~
here each distribution along an index is correlated with the previous~~
point along that index.
Author: Lonnie Chrisman, Ph.D.~
Lumina Decision Systems~
~
With contributions by:~
John Bowers, US FDA.~
Max Henrion, Lumina Decision Systems
Date: Fri, Aug 01, 2003 7:12 PM
Saveauthor: Lonnie
Savedate: Thu, Jul 26, 2007 8:47 PM
Defaultsize: 48,24
Nodelocation: 344,64,0
Nodesize: 60,32
Nodeinfo: 1,1,1,1,1,1,0,0,0,0
Diagstate: 1,42,10,649,1076,17
Windstate: 2,401,199,483,316
Function Regressiondist( Y : Numeric[I] ; B : Numeric[I,K] ; I,K : In~~
dex; C : optional Numeric[K] ;~
S : optional Numeric[I,Run] ;~
singleSampleMethod : optional hidden scalar )
Title: RegressionDist(Y,B,I,K)
Description: RegressionDist returns linear regression coefficients as ~~
a distribution.~
~
Suppose you have data where Y was produced as:~
Y = Sum( C*B, K ) + Normal(0,S)~
~
S is the measurement noise. You have the data (B[I,K] and Y[I]). Yo~~
u might or might not know the measurement noise S. So you perform a ~~
linear regression to obtain an estimate of C. Because your estimate ~~
is obtained from a finite amount of data, your estimate of C is itsel~~
f uncertain. This function returns the coefficients C as a distribut~~
ion (i.e., in Sample mode, it returns a sampling of coefficients inde~~
xed by Run and K), reflecting the uncertainty in the estimation of th~~
ese parameters.~
~
If you know the noise level S in advance, then you can use historical~~
data as a starting point for building a predictive model of Y, as fo~~
llows:~
~
{ Your model of the dependent variables: }~
Variable Y := your historical dependent data, indexed by I~
Variable B := your historical independent data, indexed by I,K~
Variable X := { indexed by K. Maybe others. Possibly uncertain }~
Variable S := { the known noise level }~
Chance C := RegressionDist(Y,B,I,K)~
Variable Predicted_Y := Sum(C*X,K) + Normal(0,S)~
~
If you don't know the noise level, then you need to estimate it. You'~~
ll need it for the normal term of Predicted_Y anyway, and you'll need~~
to do a regression to find it. So you can pass these optional param~~
eters into RegressionDist. The last three lines above become:~
Variable E_C := Regression(Y,B,I,K)~
Variable S := RegressionNoise( Y,B,I,K,E_C )~
Chance C := RegressionDist(Y,B,I,K,E_C)~
Variable Predicted_Y := Sum(C*X,K) + Normal(0,S)~
~
If you use RegressionNoise to compute S, you should use Mid(Regressio~~
nNoise(...)) for the S parameter. However, when computing S for your~~
prediction, don't RegressionNoise in context. Better is if you don'~~
t know the measurement noise in advance, don't supply it as a paramet~~
er.
Definition: if IsNotSpecified(C) Then C := Regression(Y,B,I,K);~
If IsNotSpecified(S) Then S := Mid(RegressionNoise(Y,B,I,K,C))~
Else S:=Mean(S);~
Index K2 := K;~
var cv := Invert( Sum( B*B[K=K2]/S^2,I),K,K2);~
cv := Average( [cv,Transpose(cv,K,K2)]); {for round off errs}~
~
Gaussian(C,cv,K,K2,singleSampleMethod:singleSampleMethod)
Nodelocation: 152,680,1
Nodesize: 112,20
Windstate: 2,115,1,561,882
Paramnames: Y,B,I,K,C,S
Function Gaussian(meanVec : numeric[I],covar : numeric[I,J]; I,J:Index~~
Type ;~
Over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar )
Title: Gaussian( m, cv, i, j )
Description: A multi-variate Gaussian distribution based on a mean vec~~
tor and covariance matrix. The covariance matrix must symmetric and ~~
positive-definite. The meanVec is indexed by I. The covariance matr~~
ix is 2-D, indexed by I & J. Indexes I & J should be the same length~~
.
Definition: var S := Decompose(covar,I,J);~
var U := ifall J then 0 else 0;~
var Z := Normal(U,1,singleSampleMethod:singleSampleMethod);~
sum( S*Z,J ) + meanVec
Nodelocation: 112,72,1
Nodesize: 80,16
Windstate: 2,96,299,486,314
Paramnames: meanVec,covar,I,J,Over
Function Dirichlet(alpha : Numeric[I]; I:IndexType ;~
Over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar)
Title: Dirichlet ( a, i )
Description: A Dirichlet distribution with parameters alpha_i>0~
Each sample of a Dirichlet distribution produces a random vector whos~~
e elements sum to 1. It is commonly used to represent second order p~~
robability information.~
~
The Dirichlet distribution has a density given by ~
k * Product( X^(alpha-1), I)~
where k is a normalization factor equal to~
GammaFn( sum(alpha,I )) / Sum(GammaFn(alpha),I)~
~
The parameters, alpha, can be interpreted as observation counts. The~~
mean is given by the relative values of alpha (normalized to 1), but~~
the variance narrows as the alphas get larger, just as your confiden~~
ce in a distribution would narrow as you get more samples.~
~
The Dirichlet lends itself to easy Bayesian updating. If you have a ~~
prior of alpha0, and you observe N
Definition: var a:=Gamma(alpha,singleSampleMethod:singleSampleMethod);~~
~
a/sum(a,I)
Nodelocation: 272,120,1
Nodesize: 58,16
Windstate: 2,26,18,624,485
Paramnames: alpha,I,Over
Function Binormal(MeanVec :numeric[I]; Sdeviations : positive[I]; I:In~~
dexType; correlationCoef : numeric atomic;~
Over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar)
Title: BiNormal (m, s, i, c )
Description: A 2-D Normal (or Bi-variate Gaussian) distribution with t~~
he indicated individual standard deviations (>0) and the indicated co~~
rrelation coefficient. The index, I, must have exactly 2 elements, S~~
deviations must be indexed by I.
Definition: if size(I)<>2 then ~
Error("Index to BiNormal must have 2 elements")~
else begin~
var s := product(Sdeviations,I) * correlationCoef;~
Index J:=CopyIndex(I);~
Gaussian( meanVec, If I<>J Then s else Sdeviations^2, I,J,~
singleSampleMethod: singleSampleMethod )~
end
Nodelocation: 288,72,1
Nodesize: 78,16
Windstate: 2,2,24,525,540
Paramnames: MeanVec,Sdeviations,I,correlationCoef,Over
Function Multinomial(N:Positive ; theta:NonNegative ; I : IndexType;~
Over : ... optional atomic ;~
singleSampleMethod : hidden optional scalar )
Title: Multinomial (n, theta, i )
Description: Returns the Multinomial Distribution.~
~
The multinomial distribution is a generalization of the Binomial dist~~
ribution to N possible outcomes. For example, if you were to roll a ~~
fair die N times, an outcome would be the number of times each of the~~
six numbers appears. Theta would be the probability of each outcome~~
, where sum(theta,I)=1, and index I is the list of possible outcome. ~~
If theta doesn't sum to 1, it is normalized.~
~
Each sample is a vector indexed by I indicating the number of times t~~
he corresponding outcome (die number) occurred during that sample poi~~
nt. Each sample will have the property that sum( result, I ) = N.
Definition: var z := n;~
var k := size(I);~
~
var j:=cumulate(1,I) in I do begin~
Index I2 := j..k;~
var theta2 := Slice(theta,I,I2); /* unnormalized sub-process */~
var p := theta2/sum(theta2, I2);~
p := if IsNan(p) then 0 else p;~
var xj := Binomial(z,p[I2=j],~
singleSampleMethod:singleSampleMethod);~~
~
z := z - xj;~
xj~
end~
Nodelocation: 117,120,1
Nodesize: 85,16
Windstate: 2,75,167,476,522
Paramnames: N,theta,I,Over
Function Correlate_dists(dists : Context[I,RunIndex] ; rankcorrs : num~~
eric array[I,J] ; ~
I,J : IndexType;~
RunIndex : optional Index = Run )
Title: Correlate Dists (d, rc, i, j )
Description: Reorders the samples in dists so as to match the desired ~~
rank correlations between distributions as closely as possible. Rank~~
Corrs must be positive definite, and the diagonal should contain all ~~
ones.~
~
The result will be distributions having the same margins as the origi~~
nal input, but with rank correlations close to those of the rankcorrs~~
matrix.
Definition: if not IsSampleEvalMode and Handle(RunIndex)=Handle(Run) T~~
hen~
dists {Mid mode}~
Else begin~
var u := if Handle(RunIndex)=Handle(run) ~
Then Sample(Gaussian(0,rankcorrs,I,J))~
Else Random(Gaussian(0,rankcorrs,I,J),Over:RunIndex);~
var dsort := sortIndex(dists,RunIndex);~
var urank := Rank(u,RunIndex);~
dists[RunIndex=dsort[RunIndex=urank]]~
end
Nodelocation: 136,336,1
Nodesize: 100,16
Windstate: 2,301,193,494,399
Paramnames: dists,rankcorrs,I,J,RunIndex
Function Correlate_with( S, ref : Context[RunIndex] ; rc : scalar ; ~
RunIndex : optional Index = Run )
Title: Correlate With (s, ref, rc )
Description: Reorders the samples of S so that the result is correlate~~
d with the reference sample with a rank correlation close to rankcorr~~
. ~
~
Example: To generate a logNormal distribution that is highly correlat~~
ed with Ch1, use, e.g.,: Correlate_With( LogNormal(2,3), Ch1, 0.8 )~
~
Note: This achieves a given unweighted rank correlation. If you have~~
a non-default SampleWeighting of points, the weighted rank correlato~~
n may differ.
Definition: if IsSampleEvalMode or Handle(runIndex)<>Handle(Run) Then ~~
begin~
Index q := 1..2;~
var u := If Handle(RunIndex)=Handle(Run) ~
Then binormal( 0, 1, q, rc )~
Else Random(binormal(0,1,q,rc),Over:RunIndex);~
var rrank := Rank(ref,RunIndex);~
var u1sort := sortIndex(u[q=1],RunIndex);~
var u2rank := Rank(u[q=2],RunIndex);~
var ssort := sortIndex(S,RunIndex);~
S[RunIndex=ssort[RunIndex=u2rank[RunIndex=~
u1sort[Ru~~
nIndex=rrank]]]]~
end ~
else {mid mode}~
S
Nodelocation: 128,256,1
Nodesize: 96,16
Windstate: 2,205,170,545,485
Paramnames: S,ref,rc,RunIndex
Function Uniformspherical(I : IndexType ; R : optional Numeric[I] ;~
Over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar )
Title: Uniform Spherical (i, r )
Description: Generates points uniformly on a sphere (or circle or hype~~
rsphere).~
Each sample generated is indexed by I -- so if I has 3 elements, the ~~
points will lie on a sphere.~
~
The mid value is a bit strange here since there isn't really a median~~
that lies on the sphere. Obviously the center of the sphere is the ~~
middle value, but that isn't in the allowable range. So, an arbitrar~~
y point on the sphere is used.
Definition: if IsNotSpecified(R) then R:=1;~
var u := Normal(0,1,over:I,~
singleSampleMethod:singleSampleMethod); ~
var d := sqrt( sum(u^2,I) );~
if d=0 then Array(I,R/sqrt(size(I))) else r*u/d
Nodelocation: 328,168,1
Nodesize: 86,16
Windstate: 2,151,227,476,424
Paramnames: I,R,Over
Function Multiuniform(corr : Numeric[I,J] ; I,J : IndexType ; lb,ub : ~~
optional Numeric[I,J] ;~
Over : ... optional atomic ;~
singleSampleMethod : hidden optional scalar )
Title: MultiUniform ( c, i, j, lb, ub )
Description: The multi-variate uniform distribution.~
Generates vector samples (indexed by I) such that each component has ~~
a uniform marginal distribution, and such that each component have th~~
e pair-wise correlations given by corr. Indexes I and J must have th~~
e same number of elements, corr needs to be symmetric and must obey a~~
certain semidefinite condition (namely that the transformed matrix [~~
2*sin(30*cov) ] is positive semidefinite. In most cases, this rough~~
ly the same as corr being, or not being, positive semidefinite). Lb ~~
and ub can be used to specify upper and lower bounds, either for all ~~
components, or individually if these bounds are indexed by I. If lb ~~
& ub are omitted, each component will have marginal Uniform(0,1).~
~
The correlation specified in corr is true sample correlation - not ra~~
nk correlation. ~
~
The transformation here is based on:~
* Falk, M. (1999), "A simple approach to the generation of uniformly ~~
distributed random variables with prescribed correlations," Comm. in ~~
Stats - Simulation and Computation 28: 785-791.
Definition: if IsNotSpecified(lb) then lb:=0;~
if IsNotSpecified(ub) then ub := 1;~
var R := if I=J then 1 else 2*sin(30*corr);~
var g := Gaussian(0,R,I,J,~
singleSampleMethod:singleSampleMethod);~~
~
Cumnormal( g ) * (ub-lb) + lb
Nodelocation: 132,168,1
Nodesize: 100,16
Windstate: 2,67,106,608,611
Paramnames: corr,I,J,lb,ub,Over
Module Depricated_multi_var
Title: Depricated multi-variate stuff
Description: Functions found in this module are here for legacy reason~~
s. They existed in older versions of the Multivariate library, but h~~
ave been become obsolete for whatever reason.
Author: Lonnie
Date: Mon, Apr 30, 2007 3:49 PM
Defaultsize: 48,24
Nodelocation: 80,888,1
Nodesize: 56,32
Function Samplecovariance(X ; I : Index ; J : optional Index ; R : In~~
dex)
Title: Sample Covariance
Description: This function is obsolete. In Analytica 4.0, the builtin~~
function Variance can be used to compute a covariance matrix. The e~~
quivalent of this function would be: Variance( X, R, CoVarDim:I, CoV~~
arDim2:J ).~
~
Returns a covariance matrix based on the sampled data, X, indexed by ~~
I and R. (I is the dimensionality of X, R corresponds to the samples~~
). The result will be indexed by I and J -- supply J to be the same ~~
length as I.~
~
Note that the mean is simply Average(X,R), and doen't warrant a separ~~
ate function.
Definition: var I2 := if IsNotSpecified(J) ~
Then (Index K/((identifier of I)&"2") := I do VarTerm(K~~
)) ~
Else VarTerm(J);~
var Z:=X-Average(X,R);~
var Zt := Z[@I=@I2];~
Sum(Z*Zt,R)/(size(R)-1)
Nodelocation: 80,48,1
Nodesize: 48,24
Windstate: 2,222,299,476,297
Paramnames: X,I,J,R
Function Samplecorrelation(X : array[I,R] ; I,J,R : IndexType)
Title: sample correlation
Description: This function is obsolete. A covariance matrix can be co~~
mputed in Analytica 4.0+ using the built-in function Correlation. Th~~
e equivalent of this function is Correlation(X,X[@I=@J],R).~
~
Returns a correlation matrix based on data in X, where each data poin~~
t is a vector indexed by I, and the entries in the correlation matrix~~
are the pair-wise correlations of the columns of data. A second ind~~
ex, J, of size identical to I, is required in order to index the 2-di~~
mensional result.
Definition: var z:=x-average(x,R);~
var zt := slice(z,I,cumulate(1,J));~
sum(z*zt,R) / sqrt(sum(z^2,R) * sum(zt^2,R))~
Nodelocation: 208,48,1
Nodesize: 48,24
Windstate: 2,70,24,523,377
Paramnames: X,I,J,R
Close Depricated_multi_var
Text Te1
Description: Parametric Multivariate Distributions
Nodelocation: 160,40,-1
Nodesize: 136,12
Text Te2
Description: Creating an array of mutually correlated distributions:
Nodelocation: 232,312,-1
Nodesize: 200,16
Text Te3
Description: Creating a single univariate distribution correlated wit~~
h another existing dist:
Nodelocation: 296,224,-1
Nodesize: 268,12
Function Normal_correl(m, s, r, y: Numeric ;~
over : optional atomic ;~
singleSampleMethod : optional hidden scalar )
Title: Normal_correl(m, s, r, y)
Description: Generates a normal distribution with mean m, standard dev~~
iation s, and correlation r with normally distributed value y. In a ~~
deterministic context, it will return m.~
~
If y is not normally distributed, the result will also not be normal,~~
and the correlation will be approximate. It generalizes appropriatel~~
y if any of the parameters are arrays:The result array will have the ~~
union of the indexes of the parameters.
Definition: IF r<-1 OR r>1 THEN Error('Correlation parameter r in func~~
tion Normal_correl(m, s, r, y) is outside the expected range [-1, 1].~~
');~
IFOnly IsSampleEvalMode ~
THEN m + s * (Sqrt(1-r^2) ~
* Normal(Sameindexes( 0, m ), Sameindexes( 1~~
, s ),~
singleSampleMethod:singleSampl~~
eMethod ) ~
+ r * (y - Mean(y))/Sdeviation(y))~
ELSE m
Nodelocation: 352,256,1
Nodesize: 108,16
Windstate: 2,102,90,503,416
Paramnames: m,s,r,y,over
Module Multivariate_interna
Title: Multivariate Internal Functions
Author: Lonnie
Date: Tue, May 01, 2007 9:29 PM
Defaultsize: 48,24
Nodelocation: 200,888,1
Nodesize: 52,32
Function Sameindexes(x, y)
Title: SameIndexes(x,y)
Description: Returns an array with the same indexes as y, and value x ~~
in each cell.
Definition: IF y=y THEN x ELSE x
Nodelocation: 120,64,1
Nodesize: 80,20
Paramnames: x,y
Close Multivariate_interna
Function Multinormal(m, s: Numeric; cm: ArrayType[i, j]; i , j: IndexT~~
ype ;~
Over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar )
Title: Multinormal(m,s,c,i,j)
Description: A multi-variate normal (or Gaussian) distribution with me~~
an m, standard deviation s, and correlation matrix cm. m and s may ~~
be scalar or indexed by i. cm must be symmetric, positive-definite, a~~
nd indexed by i & j, which must be the same length.~
~
Multinormal uses a correlation matrix. Compare with Gaussian, which ~~
also defines a multi-variate normal but which uses a covariance matri~~
x.
Definition: Gaussian(m,cm*s*s[@i=@j],i,j,over,singleSampleMethod)
Nodelocation: 472,72,1
Nodesize: 84,16
Windstate: 2,391,248,512,343
Paramnames: m,s,cm,i,j,Over
Text Te4
Description: Reshaped distributions:
Nodelocation: 136,392,-1
Nodesize: 100,16
Function Dist_reshape(x : Numeric[R] ; newdist : all Numeric[R] ; ~
R : optional Index = Run )
Title: Dist_reshape(x, newdist)
Description: Reshapes the probability distribution of uncertain quanti~~
ty x so that it has the same marginal probability distribution (i.e, ~~
same set of sample values) as newdist, but retains the same ranks as ~~
x. Thus:~
Rank(Sample(x), Run) ~
= Rank(Sample(Reshape_dist(x, y)), Run)~
In a Mid context, it simply returns the mid value of newdist, with an~~
y indexes of x.~
~
The result retains any rank correlations that x may have with other p~~
redecessor variables. So, the rank-order correlation between a third~~
variable z and x will be the same as the rank-order correlation betw~~
een z and a reshaped version of x, i.e.~
RankCorrel(x, z) = RankCorrel(Reshape_Dist(x, y), z)~
~
The operation may optionally be applied along an index other than Run~~
.
Definition: IFOnly IsSampleEvalMode or Handle(R)<>Handle(Run) THEN BEG~~
IN~
VAR dsort := SortIndex(newdist, Run);~
VAR xranks := Rank(x, Run);~
newdist[Run = dsort[Run=xranks]]~
END~
ELSE newdist * (x=x)
Nodelocation: 152,416,1
Nodesize: 116,16
Windstate: 2,102,90,646,469
Paramnames: x,newdist,R
Text Te5
Description: Arrays with serial correlation
Nodelocation: 208,476,-1
Nodesize: 168,12
Function Normal_serial_correl(m, s, r: Numeric; i: IndexType ;~
over : ... optional atomic;~
singleSampleMethod : optional hidden scalar )
Title: Normal_serial_correl(m,s,r,i)
Description: Generates an array over index i of normal distributions w~~
ith mean m, standard deviation s, and correlation r between successiv~~
e values over index i. You can give each distribution a different m~~
ean and/or standard deviation if m and/or s are arrays indexed by i. ~~
If r is indexed by i, r[i=k] specifies the correlation between result~~
[i=k] and result[i=k-1]. (Then the first correlation, slice(r, i, 1)~~
is ignored.)
Definition: Var x := Normal(0, 1,singleSampleMethod:singleSampleMethod~~
);~
(FOR j := i DO ~
x := Normal_correl( 0, 1, r[i = j],x,~
singleSampleMethod:singleSampl~~
eMethod ) ) ~
* s + m
Nodelocation: 160,504,1
Nodesize: 120,16
Windstate: 2,353,325,540,383
Paramnames: m,s,r,i,over
Function Normal_additive_gro(x, m, s, r: Numeric; i: IndexType ;~
over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar )
Title: Normal_additive_gro(x,m,s,r,i)
Description: Adds a normally distributed percent growth g with mean m ~~
and standard deviation s to x for each value of index i. The growth g~~
for each i has serial correlation r with g for i-1.
Definition: x *( 1 + Cumulate(Normal_serial_correl(m, s, r, i,~
singleSampleMethod:singleSampleMethod), i~~
))
Nodelocation: 159,544,1
Nodesize: 119,16
Windstate: 2,102,90,519,306
Paramnames: x,m,s,r,i,over
Function Normal_compound_gro(x, m, s, r: Numeric; t: IndexType ;~
over : ... optional atomic;~
singleSampleMethod : optional hidden scalar )
Title: Normal_compound_gro(x,m,s,r,t)
Description: An array of values over time index t, starting from with ~~
value x, and with compound growth applied for each time interval, wit~~
h normal uncertainty with mean m and standard deviation s The growth~~
g for each i has correlation r with g for i-1.
Definition: x * Cumproduct(IF t = Slice(t, 1) THEN 1 ELSE Normal_seria~~
l_correl(m, s, r, t, singleSampleMethod:singleSampleMethod ) + 1, t)~~
Nodelocation: 159,584,1
Nodesize: 119,16
Windstate: 2,102,90,529,366
Paramnames: x,m,s,r,t,over
Function Dist_serial_correl(x; r; i: IndexType ;~
over : ... optional atomic;~
singleSampleMethod : optional hidden scalar )
Title: Dist_serial_correl(x,r,i)
Description: Generates an array y over index i where each y[i] has a m~~
arginal distribution identical to x, and serial rank correlation of ~~
r with y[i-1]. If x is indexed by i, each y[i] has the same margin~~
al distribution as x[i], but with samples reordered to have the speci~~
fied rank correlation r between successive values. If r is indexed b~~
y i, r[i=k] specifies the rank correlation between y[i=k] and y[i=k-1~~
]. Then the first correlation, r[i=1], is ignored.~
~
In Mid context, it returns Mid(x).~
~
Note: The result retains no probabilistic dependence on x.
Definition: Dist_reshape(Normal_serial_correl( 0, 1, r, i, singleSampl~~
eMethod:singleSampleMethod ), x)
Nodelocation: 408,504,1
Nodesize: 120,16
Windstate: 2,302,78,477,447
Paramnames: x,r,i,over
Function Dist_additive_growth(x, g, r: Numeric; i: IndexType;~
over : ... optional atomic;~
singleSampleMethod : optional hidden scalar )
Title: Dist_additive_growth(x,g,r,i)
Description: Generates an array of values over index i, with the first~~
equal to x, and successive values adding an uncertain growth with pr~~
obability distribution g, and serial correlation r between growth[i =~~
k] and growth[i=k-1]. x, g, and r each may be indexed by i if you w~~
ant them to vary over i.
Definition: x + Cumulate(Dist_serial_correl( g, r, i, singleSampleMeth~~
od : singleSampleMethod), i)
Nodelocation: 407,544,1
Nodesize: 119,16
Windstate: 2,102,90,506,300
Paramnames: x,g,r,i,over
Function Dist_compound_growth(x, g, r; i: IndexType ;~
over : ... optional atomic ;~
singleSampleMethod : optional hidden scalar )
Title: Dist_compound_growth(x,g,r,i)
Description: Starts with x and applies a compound growth g for each va~~
lue of index i. The growth g for each i has correlation r with g for ~~
i-1.
Definition: x * Cumproduct(~
IF i = Slice(i, 1) THEN 1 ~
ELSE (Dist_serial_correl( g, r, i, ~
singleSampleMethod:singleSampleMe~~
thod ) + 1)~
, i)
Nodelocation: 407,584,1
Nodesize: 119,16
Windstate: 2,102,90,489,307
Paramnames: x,g,r,i,over
Text Te6
Description: Distributions on Linear Regression coefficients
Nodelocation: 296,632,-1
Nodesize: 256,12
Function Regressionnoise( Y : Numeric[I] ; B : Numeric[I,K] ; I,K : In~~
dex; C : optional Numeric[K] )
Title: RegressionNoise(Y,B,I,K,C)
Description: When you have data, Y[I] and B[I,K], generated from an un~~
derlying model with unknown coefficients C[k] and S of the form:~
~
Y = Sum( C*B, I) + Normal(0,S)~
~
This function computes an estimate for S. ~
~
When using in conjunction with RegressionDist, it is most efficient t~~
o provide the optional parameter C to both routines, where C is the e~~
xpected value of the regression coefficients, obtained from calling R~~
egression(Y,B,I,K). Doing so avoids an unnecessary call to the built~~
in Regression function.
Definition: if IsNotSpecified(C) Then C := Regression(Y,B,I,K);~
Var resid := Y - Sum(C*B,K);~
sqrt( Sum(resid^2,I) / (size(I)-size(K)) );~
Nodelocation: 384,680,1
Nodesize: 104,20
Windstate: 2,332,211,498,542
Paramnames: Y,B,I,K,C
Function Regressionfitprob( Y : Numeric[I] ; B : Numeric[I,K] ; I,K : ~~
Index; C : optional Numeric[K] ; ~
S : optional Numeric[I] )
Title: RegressionFitProb(Y,B,I,K,C)
Description: Once you've obtained regression coefficients C (indexed b~~
y K) by calling the Regression function, this function returns the pr~~
obability that a fit this poor would occur by chance, given the assum~~
ption that the data was generated by a process of the form:~
~
Y = Sum( C*B,K) + Normal(0,S)~
~
If this result is very close to zero, it probably indicates that the ~~
assumption of linearity is bad. If it is very close to one, then it ~~
validates the assumption of linearity.~
~
This is not a distribution function - it does not return a sample whe~~
n evaluated in Sample mode. However, it does complement the multivar~~
iate RegressionDist function also included in this library.~
~
To use, first call the Regression function, then you must either know~~
the measurement knows a priori, or obtain it using the RegressionNoi~~
se function.~
~
Var E_C := Regression(Y,B,I,K);~
Var S := RegressionNoise(Y,B,I,K,C);~
Var PrThisPoor := RegressionFitProb(Y,B,I,K,E_C,S)
Definition: var resid := Y - sum(C*B,K);~
var n := size(I);~
var chi2 := sum( resid^2 / Mean(S)^2, I);~
GammaI( n/2 - 1, chi2/2 )
Nodelocation: 152,744,1
Nodesize: 112,20
Windstate: 2,287,69,586,548
Paramnames: Y,B,I,K,C,S
Close Multivariate_distrib
Chance Simulated_sp
Title: Simulated SP
Definition: Gaussian( Mean_sp, Cov_mat, Stock, Stock2 )
Nodelocation: 208,304,1
Nodesize: 48,24
Valuestate: 2,236,144,552,403,1,SAMP
Reformval: [Undefined,Undefined,Undefined,Undefined,Undefined,1]
{!40000|Att_xrole: -2}
{!40000|Att_yrole: -3}
{!40000|Att_coordinateindex: Stock}
Function Some_statistic( X : ContextSamp[I] ; I : optional Index = Run~~
)
Title: some statistic
Definition: Variance(X,I) / Mean(X,I)
Nodelocation: 352,160,1
Nodesize: 48,24
Windstate: 2,527,133,476,224
Paramnames: X,I
Variable Va2
Definition: Some_statistic(Stock_price,Trading_date)
Nodelocation: 328,248,1
Nodesize: 48,24
Chance Va4
Definition: random(Normal(0,1),over:stock)
Nodelocation: 72,304,1
Nodesize: 48,24
Valuestate: 2,267,242,416,303,0,MIDM
Close Historical_data
Chance X
Title: X
Definition: lognormal(mean:5,stddev:3)
Nodelocation: 208,48,1
Nodesize: 48,24
Valuestate: 2,551,37,405,457,0,PDFP
Graphsetup: Statsselect:[1, 1, 1, 1, 1, 1, 1, 1 ]
Variable Mean_x
Title: mean x
Definition: mean(X)
Nodelocation: 328,48,1
Nodesize: 48,24
Valuestate: 2,72,291,416,303,0,SMPL
Variable Y
Title: Y
Definition: X+1
Nodelocation: 200,120,1
Nodesize: 48,24
Valuestate: 2,136,146,416,303,0,SAMP
Variable Variance_x
Title: variance x
Definition: variance(X)
Nodelocation: 328,112,1
Nodesize: 48,24
Valuestate: 2,152,242,416,303,0,MIDM
Variable Sample_x
Title: sample x
Definition: sample(X)
Nodelocation: 328,176,1
Nodesize: 48,24
Valuestate: 2,270,191,416,303,0,MIDM
Variable Min_x
Title: min x
Definition: Min( Sample(X), Run )
Nodelocation: 328,240,1
Nodesize: 48,24
Valuestate: 2,466,46,416,303,0,MIDM
Variable Average_x
Title: average x
Definition: average( Sample(x), Run )
Nodelocation: 440,48,1
Nodesize: 48,24
Valuestate: 2,517,150,416,303,0,MIDM
Variable Histo_x
Title: histo x
Definition: Pdf(X)
Nodelocation: 208,192,1
Nodesize: 48,24
Valuestate: 2,87,115,560,494,1,MIDM
Reformval: [Sys_localindex('STEP'),Densityindex]
{!40000|Att_coordinateindex: Densityindex}
Chance Z
Title: Z
Definition: X^2 + normal(0,10)
Nodelocation: 72,120,1
Nodesize: 48,24
Valuestate: 2,156,274,524,303,1,PDFP
Variable Correl_z_vs_x
Title: correl z vs x
Definition: correlation(x,z)
Nodelocation: 72,184,1
Nodesize: 48,24
Variable Cov
Title: cov
Definition: covariance(x,z)
Nodelocation: 72,240,1
Nodesize: 48,24
Valuestate: 2,246,229,416,303,0,MIDM
Variable Va1
Definition: Some_statistic(Z)
Nodelocation: 192,256,1
Nodesize: 48,24
Module Discrete_vs_continuo
Title: Discrete vs continuous
Defaultsize: 48,24
Nodelocation: 448,144,1
Nodesize: 48,32
Diagstate: 1,94,64,550,300,17
Chance Count
Title: count
Definition: poisson(100)
Nodelocation: 88,64,1
Nodesize: 48,24
Valuestate: 2,94,66,762,303,1,PDFP
Domain: Domcontinuous
Variable Histo_count
Title: histo count
Definition: pdf(Count,discrete:true)
Nodelocation: 232,64,1
Nodesize: 48,24
Valuestate: 2,366,45,523,453,0,SMPL
Reformval: [Sys_localindex('STEP'),Densityindex]
Chance A
Title: A
Definition: Probtable(Self)(~
0.3,0.2,0.5)
Nodelocation: 88,144,1
Nodesize: 48,24
Valuestate: 2,449,123,366,455,0,SAMP
Domain: ['item 1','item 2','item 3']
Variable Va3
Definition: frequency( A, poss_A )
Nodelocation: 216,144,1
Nodesize: 48,24
Valuestate: 2,435,240,416,303,0,MIDM
Index Poss_a
Title: Poss A
Definition: ['item 1','item 2','item 3']
Nodelocation: 88,208,1
Nodesize: 48,24
Close Discrete_vs_continuo
Module Weighting_of_samples
Title: Weighting of samples
Defaultsize: 48,24
Nodelocation: 432,240,1
Nodesize: 48,24
Diagstate: 1,156,94,741,445,17
Chance U
Title: U
Definition: uniform(-1,1)
Nodelocation: 104,64,1
Nodesize: 48,24
Chance V
Title: V
Definition: uniform(-1,1)
Nodelocation: 104,136,1
Nodesize: 48,24
Valuestate: 2,168,178,416,303,1,SAMP
Xyexpr: U
Variable V1
Title: V
Definition: (U + V) / (abs(U - V) + 1)
Nodelocation: 224,96,1
Nodesize: 48,24
Valuestate: 2,217,157,650,431,1,PDFP
Graphsetup: {!40000|Att_contlinestyle Graph_pdf_valdim:1}
Reformval: [Undefined,Result_type,Undefined,Undefined,1]
Variable Wt
Title: wt
Definition: Table(Result_type)(1, U<=0 or V <=0)
Nodelocation: 224,168,1
Nodesize: 48,24
Valuestate: 2,184,194,279,400,0,SAMP
Variable Posterior_mean
Title: posterior mean
Definition: Mean(V1, w:Wt )
Nodelocation: 344,168,1
Nodesize: 48,24
Variable Posterior_variance
Title: posterior variance
Definition: variance(V1, w:Wt)
Nodelocation: 344,224,1
Nodesize: 48,24
Decision Result_type
Title: Result type
Definition: Choice( Self, 0 )
Nodelocation: 96,208,1
Nodesize: 48,24
Domain: ['Prior','Posterior']
{!40000|Att_previndexvalue: ['Prior','Posterior']}
Chance Ch1
Definition: Normal( sm, ssd, over:Pass )
Nodelocation: 456,136,1
Nodesize: 48,24
Valuestate: 2,284,256,416,303,1,PDFP
Reformval: [Undefined,Pass,Undefined,Undefined,1]
Objective Cost
Title: Cost
Definition: exp(Ch1/10)
Nodelocation: 456,208,1
Nodesize: 48,24
Valuestate: 2,90,292,416,303,1,STAT
Graphsetup: Statsselect:[1, 1, 1, 1, 1, 1, 0, 0 ]
Reformval: [Undefined,Sys_localindex('STATISTICS'),Undefined,0]
Xyexpr: Ch1
Variable Adj_wt
Title: adj wt
Definition: Dens_Normal(Ch1,10,5) / Dens_Normal(Ch1,sm,ssd)
Nodelocation: 576,136,1
Nodesize: 48,24
Valuestate: 2,120,130,416,303,1,SAMP
Xyexpr: Ch1
Library Distribution_densiti
Title: Distribution Densities Library
Description: The functions in this library return the probability dens~~
ities (for continuous distributions) and probabilities (for discrete ~~
distributions) for the standard distribution functions that are built~~
into Analytica.~
~
These densities are useful for importance sampling. In importance sa~~
mpling, you can sample from a distribution different from your target~~
distribution, and then weight each point by f_target(x) / f_sample(x~~
), where f_target is the density (or prob) of your target distributio~~
n, f_sample the density (or prob) of your sample distribution.
Author: Lonnie Chrisman, Ph.D.~
Lumina Decision Systems
Date: Fri, Nov 17, 2006 8:19 AM
Saveauthor: Lonnie
Savedate: Thu, Mar 29, 2007 9:21 PM
Defaultsize: 48,24
Nodelocation: 336,40,0
Nodesize: 60,32
Nodeinfo: 1,1,1,1,1,1,0,0,0,0
Diagstate: 1,20,25,557,391,17
Windstate: 2,102,90,582,409
Fontstyle: Arial, 15
Function Dens_normal(x,m,sd : numeric)
Title: Dens_Normal(x,m,sd)
Description: The density of a uniform distribution
Definition: 1/(sd*sqrt(2*Pi) ) * exp( - 0.5 * ( (x-m)/sd)^2 )
Nodelocation: 96,120,1
Nodesize: 60,24
Windstate: 2,105,359,476,224
Paramnames: x,m,sd
Function Dens_uniform_(x,min,max:Numeric ; integer : optional boolean ~~
= false)
Title: Dens_Uniform~
(x,min,max)
Description: The probability density of x in a uniform distribution. ~~
If the parameter integer is supplied and specified as false, then the~~
probability of a discrete uniform distribution is returned.
Definition: if integer then 1/(floor(max)-ceil(min)+1)~
else 1/(max-min)
Nodelocation: 224,120,1
Nodesize: 60,24
Paramnames: x,min,max,integer
Function Dens_beta(x,a,b : numeric ; lower : optional=0, upper : optio~~
nal = 1)
Title: Dens_Beta~
(x,a,b)
Description: The density of a beta distribution at x.
Definition: x := (x-lower) / (upper-lower);~
1/Betafn( a,b) * x^(a-1) * (1-x)^(b-1) / (upper-lower)
Nodelocation: 224,56,1
Nodesize: 60,24
Paramnames: x,a,b,lower,upper
Function Prob_bernoulli(x ; p : NonNegative)
Title: Prob_Bernoulli~
(x,p)
Description: The probability of a Bernoulli distribution with paramete~~
r p at x. Since a Bernoulli sample consists of 0s and 1s only, the p~~
robability is zero everywhere except at x=0 and x=1.
Definition: if x=0 then (1-p)~
else if x=1 then p~
else 0
Nodelocation: 96,56,1
Nodesize: 60,24
Paramnames: x,p
Function Prob_binomial(x, n,p : NonNegative)
Title: Prob_Binomial(x,n,p)
Description: The probability that x positive events occur in a Binomia~~
l sample of size n drawn, where the independent probability of drawin~~
g a positive sample in each draw is p.
Definition: if x<0 or x>n then 0 ~
else~
Combinations(x,n) * p^x * (1-p)^(n-x)
Nodelocation: 352,56,1
Nodesize: 60,24
Paramnames: x,n,p
Function Prob_chancedist(x ; p:NonNegative[I] ; A : Array[I] ; I)
Title: Prob_ChanceDist~
(x,P,A,I)
Description: Returns the probability that a sample from ChanceDist(p,A~~
,I) is x.
Definition: var j := subIndex(A,x,I);~
if IsUndef(j) then 0~
else p[I=j]
Nodelocation: 360,120,1
Nodesize: 72,24
Paramnames: x,p,A,I
Function Dens_chisquared(x ; dof : positive)
Title: Dens_ChiSquared~
(x,dof)
Description: Returns the probability at x of a chi-squared distributio~~
n with dof degrees of freedom.
Definition: 0.5^(dof/2) / Gammafn( dof/2 ) * x^(dof/2-1) * exp(-0.5 * ~~
x)
Nodelocation: 112,184,1
Nodesize: 72,24
Paramnames: x,dof
Function Dens_exponential(x,rate : numeric)
Title: Dens_Exponential~
(x,rate)
Description: The density of an exponential distribution with the given~~
rate parameter at x.
Definition: rate * exp(-rate * x)
Nodelocation: 264,184,1
Nodesize: 72,24
Paramnames: x,rate
Function Dens_gamma(x : Nonnegative ; alpha : positive ; beta : option~~
al = 1 )
Title: Dens_Gamma~
(x,a,b)
Description: The probability density of a gamma(alpha,beta) distributi~~
on at x.
Definition: beta^(-alpha) * x^(alpha-1) * exp(-x/beta) / gammafn(alph~~
a)
Nodelocation: 112,248,1
Nodesize: 72,24
Windstate: 2,473,89,476,218
Paramnames: x,alpha,beta
Close Distribution_densiti
Variable Vc
Title: vc
Definition: Variance(Cost)
Nodelocation: 456,288,1
Nodesize: 48,24
Valuestate: 2,44,129,416,303,0,MIDM
Reformval: [Pass,Dist_to_use]
Index Year
Title: Year
Definition: [2006,2007]
Nodelocation: 96,288,1
Nodesize: 48,24
Index Quarter
Title: Quarter
Definition: ['2006 Q1','2006 Q2','2006 Q3','2006 Q4','2007 Q1','2007 Q~~
2','2007 Q3']
Nodelocation: 200,288,1
Nodesize: 48,24
{!40000|Att_previndexvalue: ['2006 Q1','2006 Q2','2006 Q3','2006 Q4','~~
2007 Q1','2007 Q2','2007 Q3']}
Variable Revenue_by_quarter
Title: Revenue by Quarter
Definition: Table(Quarter)(~
33,44,55,66,77,55,33)
Nodelocation: 192,344,1
Nodesize: 48,24
Variable Qtoy
Title: QtoY
Definition: Table(Quarter)(~
2006,2006,2006,2006,2007,2007,2007)
Nodelocation: 328,288,1
Nodesize: 48,24
Variable Revenue_by_year
Title: Revenue by Year
Definition: Sum( Revenue_by_quarter * ( Year=Qtoy ), Quarter )
Nodelocation: 304,344,1
Nodesize: 48,24
Variable Revenue_by_year1
Title: revenue by year
Definition: Frequency( Qtoy, Year, Quarter, w: Revenue_by_quarter )
Nodelocation: 304,400,1
Nodesize: 48,24
Index Pass
Title: pass
Definition: 1..10
Nodelocation: 568,208,1
Nodesize: 48,24
{!40000|Att_previndexvalue: [1,2,3,4,5,6,7,8,9,10]}
Variable Precision_of_vc
Title: precision of vc
Definition: sdeviation(vc,pass)
Nodelocation: 576,288,1
Nodesize: 48,24
Decision Dist_to_use
Title: Dist to use
Definition: Choice(Self,0)
Nodelocation: 456,56,1
Nodesize: 48,24
Domain: ['target','sample dist']
{!40000|Att_previndexvalue: ['target','sample dist']}
Variable Sm
Title: sm
Definition: determTable(Dist_to_use)(~
10,15)
Nodelocation: 568,56,1
Nodesize: 48,24
Variable Ssd
Title: ssd
Definition: Determtable(Dist_to_use)(~
5,7)
Nodelocation: 664,56,1
Nodesize: 48,24
Close Weighting_of_samples
Close Statistical_function