Statool User s Manual

Transcription

1 Statool User s Manual Prepared by Tuan Cao with Jianzhong Zhang 7/19/04

2 Statool User s Manual Table of contents Purpose of Statool... 2 Input/Output Files for Statool...3.PDF file...3.cdf file... 3.IDF file SMP file... 3 Operand X and Y...4 Histogram Editor... 5 Toolbar PDF Mode CDF Mode... 9 Data Viewer...10 Result Z Three or more operands Correlation Unknown dep Known dep...13 Independent Operations Basic operations Max, Min Parsing Relational Operations (>,>=,<,<=) Combine Main Toolbar File, Edit, Switch menu Option, View menu Help menu Configuration Index

3 Statool Statool is a software tool for obtaining bounds on the distributions of sums, products, and various other functions of random variables, it uses the Distribution Envelop Determination (DEnv) algorithm. Figure 1: The main layout screen. Statool takes two input random variables as X and Y and the relationship between them as unknown dependent, known dependent or independent. It then will obtain bounds on the distribution of various functions. There are the basic arithmetic (+,-,*,/), max, min, relational operators and it can also take an input function of X and Y (Parsing). The result will be displayed Z in the third sub-window. 2

4 Input/Output Files Files are a primary way to exchange and save the data for this software. Graphs are used to show probability distribution functions (PDFs) and cumulative distribution functions (CDFs). Text-based data lists can show the exact information representing a random variable. In this software, 4 kinds of files are used to describe random variables. One is called a probability distribution file (.PDF), which is a list of intervals and probabilities, and discretizes a probability density function. The second is called a cumulative distribution file (.CDF), which is used to describe the envelopes of the cumulative distribution. The third kind is called an intermediate distribution file (.IDF), which is like a.pdf except that the intervals can overlap. Although its interval can overlap, the sum of their probabilities should be equal to 1. Otherwise, the file is invalid. The last one is the sample data file (.SMP). It is a special type of file. It is used to show the probability distribution of sample data. Only.IDF and.smp files are used to input or output distributions to or from disk..pdf and.cdf files are used internally. Output displays are plotted based on these 2 types of files. Both.IDF and.smp files are text based. They are composed of 2 parts: a control line and data lines. The control line is the first line and includes the number of intervals in the file. This is also the number of data lines in the file. Every data line includes 3 items: low bound of an interval, high bound, and probability for this interval. The comma is used as a separator. The following file gives an example IDF file: 4,0 0,0.25, ,0.50, ,0.75, ,1.0,0.25 This file describes the distribution of a variable consisting of 4 bars from 0 to 1. A sample file is a special.idf. In this type of file, the low bound of every interval is set to equal the high bound. The third number on the line states the probability a sample takes that value. Two different ways are used to input data in Statool. One is to load the distribution of a random variable from a file. The other is to edit the distribution of a variable. There are two ways to output the result of an operation. One is to draw the graph of the variable. This is convenient for seeing the result immediately. The other way is to save data as an.idf or.smp file. These two files were discussed in the previous paragraph. There are 2 types of graph for showing the data: histograms for PDF format and envelopes for CDF format, which show the space of possible cumulative probability functions of a derived random variable. 3

5 Operand X and Y To input a random variable X(Y): Left or Right click on the X(Y) sub-window and it will bring up a menu of options to choose from (Figure 2). Figure 2: Menu for X and Y New X(Y): This will create a new file for X(Y). It will bring up a Histogram Editor (Figure 3) and the default mode is cumulative distribution function (CDF). To edit the file, change it to PDF mode. Open X(Y): Click on this to open a previously saved X(Y). Save X(Y) As: Save the current X(Y) with a specific name. Edit X(Y): Bring up the Histogram Editor (Figure 3) for modifying values of current X(Y). Clear X(Y): Clear the current X(Y). Hint: Try clicking on new X(Y). 4

6 Histogram Editor Figure 3: Histogram Editor for X (CDF mode) In this window there are four main menus: File: This menu has five options: New: Make a new file for X(Y) (change to PDF mode). Open: Open an existing file for X(Y). Save: Save the current file for X(Y). Save as: Save the current file for X(Y) with a specific name. Print: Print the current window. Edit: This menu has four options: No. of Bars(PDF Mode): (Figure 4) To enter the number of bars wanted on the histogram. PDF Bounds(PDF Mode): (Figure 5) To set the upper and lower bound for the histogram. Single Bar(PDF Mode): (Figure 6) To edit any bar by inputting it s lower and upper bound and probability. Clear Window: Clear the screen. 5

7 Figure 4: Type-in box for number of bars Figure 5: Type-in box for PDF Bounds Figure 6: Type-in box for each interval. View: This menu has nine options: PDF mode: (Figure 9) View the file as a histogram (bars). CDF mode: (Figure 10) View the file as left and right envelopes (staircase). Grid On(default): Show grid on the graph in CDF mode. Grid Off: Don t show grid on the graph in CDF mode. Color(default): Show colors on graphs of both modes. Black and White: Show only black and white on graphs of both modes. Data: (Figure 12) Open up a data viewer to view the exact data for X, Y or Z. Font Size: Select from three different size of font (Big, Medium(default), Small). Draw by Scale: (Figure 7) Change the start and end value of x-axis and number of intervals for the CDF mode. 6

8 Figure 7: Draw by Scale Help: This menu has two options: Content: Open up a help file. About Statool: (Figure 8) Information about Statool. Figure 8: About Statool Hint: To edit, try converting to histogram form: click View and PDF Mode. 7

9 PDF Mode To change to PDF mode: click on View and PDF mode. Figure 9: Histogram Editor for X (PDF mode). Divide Bar: Doubles the number of histogram by dividing each one in half. Merge Bar: Half the number of histogram. Incr Bar: Add a bar to the histogram. Decr Bar: Remove a bar to the histogram. To change the height of each bar: Left click above the bar causes the bar to rise; Left click below the top of the bar causes it to shorten. To change the width of each bar: Right click above the bar causes the bar to widen; Right click below the top of the bar causes it to become narrower. 8

10 CDF Mode Figure 10: CDF Mode(Grid off) Figure 11: Expand To Change to CDF Mode: click on View and then CDF Mode. Expand: (Figure 11) This will let you rediscretize CDF, it will let you decrease the number of intervals. Vertical Avg: At every point on the x-axis it will average the two y values. Horizontal Avg: At every point on the y-axis it will average the two x values. Fixed PT Avg: Average both vertical and horizontal points. Averaging WT: Set the weight of the left and right envelope to take average of. 9

11 Data Viewer Figure 12: Data Viewer PDF: View the exact data of X or Y. CDF: View the exact data of the lower or upper bound of X, Y, or Z. IDF: View the exact data of the file for X, Y, or Z. Cancel: Quit the Data Viewer. 10

12 Result Z Figure 13: Menu for Z Enlarge graphic: Open a new window with the result. View data: Open up the Data Viewer (Figure 12) window. Clear window: Clear the result. Edit graphic: Open up the histogram editor (Figure 12) to edit the result. Hint: Try clicking on Edit graphic. 11

13 Three or more operands This software only operates on two operands, but in real applications, there are often more than 2 operands to be calculated. For example, a+b+c, Max(a,b,c), etc. Association may be used to solve many such problems. To be able to operate on more than 2 operands, we can first operate on two of the operands and save the result. Then use that result as one operand on the next instance and so on. E.g., for a+b+c, we can first calculate a+b, and save the result to temporary variable w=a+b, then calculate w+c. However, the operation must support association and commutation. 1. Input a into X and b into Y. Add them together. 2. Save the result and input it into X. 3. Input c into Y and add them together. To extend Statool to handle cascaded operations, the following capability was added: The CDF envelopes for the result of two variables operation were converted into a marginal of a joint distribution tableau, call it w, for use in the second step of the solution process. 12

14 Correlation Correlation measures the degree of correspondence between random variables. By far the most popular correlation coefficient is called Pearson correlation, or Pearson productmoment correlation. It measures the strength of the linear relationship between two random variables. where D(X) is the variance of X and D(Y) is the variance of Y. E is the expectation function. It is well known that Pearson correlation can has the potential range Correlation values can be classified into 3 types: positive correlation, meaning there is a direct linear correlation between the R.V. s, negative correlation, meaning there is an inverse linear correlation between the R.V. s, and zero correlation, meaning there is no apparent linear relationship between the R.V. s. There are three relationships between the operands: Figure 14: Three relationships between the operands. Unknown dep: There is a correlation between the operands but it is unknown. 13

15 Known dep: There is a known correlation between the inputs. Choosing this option will bring up a window (Figure 8) to input their correlation. Figure 15: Settings for Known dependence From this window, there are 3 ways to input information about operand X and Y: known correlation range or exact value, known expectation range for XY, and known expectation and variance for X and Y. Figure 16: Prob. Matrix 14

16 Figure 17: Midpoint Alg. Figure 18: Prob. Interval Matrix There are also three more options at the bottom of the window: Prob. Matrix: (Figure 16) To input the probability of each cell in the matrix (pij s). Midpoint Alg.: (Figure 17) Each operand is averaged to one curve (each of low bound and high bound takes half weight for this curve). Then using the input correlation, two curves are calulated to get the result. Prob. Interval Matrix: Figure 18) Open up a window to input the bounds of each cell in the matrix (range for the pij s). Independent: There is no relationship between the inputs. 15

17 Operate on the Inputs Basic operations: X+Y, X-Y,X*Y,X/Y Max(X,Y): If x is a sample of X and y is a sample of Y, then a corresponding sample of Z is z=max(x,y). Define max(x,y) for intervals X and Y to be [max(low bound of X, low bound of Y), max(high bound of X, high bound of Y)]. Min(X,Y): If x is a sample of X and y is a sample of Y, then a corresponding sample of Z is z=min(x,y). Define max(x,y) for intervals X and Y to be [min(low bound of X, low bound of Y), min(high bound of X, high bound of Y)]. Parsing: Click this will bring up the expression editor (Figure 9). The parser, implemented in Statool, supports 2 random variables, named X and Y. Thus when an expression is typed in, variable names must use the symbols X and Y. The expression editor will check the input expression after the user confirms the input. If this expression is not allowed, error information will be displayed and reason also will be listed. Figure 19: Expression Editor. (Parsing) The parser also supports non-monotonic random variables. Check the corresponding box if X or Y is non-monotonic and enter the number of steps between the intervals that you want the algorithm to go through. 16

18 Relational Operations: Consider two real numbers x and y. We define the interval value to describe the relationship between x and y. The value [0,0] indicates that the relationship is false. The value [1,1] indicates the relational operation is true. The value [0,1] means the value of the relational operation is not determined or is uncertain. For interval A, A-left means the left (or low) bound of A, and A-right means the right (or high) bound. Now consider two intervals A and B. Combine: Merge the two operands together, using Figure 20 to input the weight of X and weight of Y will be (1-weight of X). Figure 20: Combine 17

19 Main Toolbar Figure 21: File menu New: Create a new IDF file for operand X or Y. Open: Open the existing file for operand X or Y. Save: Save the current result Z. Save As: Save the current result Z as a specific file. Print: Print the current operation window. Exit: Terminate the current program. Figure 22: Edit menu Edit PDF: Activate the editor window to edit the.pdf file chosen from either panel X, Y or Z. Clear Window: Clear the specific display panel. Figure 23: Switch menu Interchange X <->Y: Switch operands X and Y. Interchange X <-> Z: Switch operands X and Z. Interchange Y <-> Z: Switch operands Y and Z. 18

20 Figure 24: View menu Display Mode: Activate the property window to set the display mode for operands (Figure 14). Data: Activate the data view primary window (Figure 6). Figure 25: Display mode Figure 26: Options menu ColorSet: Activate the property window to set the display color for panels (Figure 16). Algorithm: Set the preferred algorithm to handle the linear programming problems. There are two choices: Simplex method and Transportation. 19

21 Figure 27: Color Setting Figure 28: Help menu Contents: Display help information. About Statool: Activate the property window to show the about information. Figure 29: Hint menu 20

22 Configuration File Statool has a configuration file to control the behavior of solving linear program. When the program starts, it automatically checks whether this file exists. If exist, it reads the parameters to set the control parameters of the program. Otherwise, the default parameters are used. Mainly there are two parts in this file: parameters for simplex method and parameters for transportation method. These parameters are organized as 2 sections into a configure file, called statool.cfg, which is in the same directory as that of Statool. The following is an example for this file: [Simplex] EPS= LEAVE_EPS=1.0e-16 INVERSE_EPS= MAX_LOOPNUMBER= [Transportation] Russell=0 In the section [Simplex], there are 4 parameters: EPS, LEAVE_EPS, INVERSE_EPS, and MAX_LOOPNUMBER. EPS is the minimum value so that any value less than it is considered as zero. LEAVE_EPS is used in the procedure of choosing the leave variable of simplex method. Any values less than this value are treated as zero. INVERSE_EPS is similar to LEAVE_EPS except that it is used to control calculation of inverse matrix. MAX_LOOPNUMBER is the maximum number for iterating in the simplex method. In the section [Transportation], there is only one parameter to control which kind of method is chosen to get the initial solution for transportation method. If parameter Russell is set to zero, the approach Russell is not used. 21

23 Index Averaging WT 9 CDF mode 4,9 ColorSet 19,20 Combine see Operations Correlation 13 Cumulative Distribution Function (CDF) 3 Data Viewer 10 Decr. bar 8 DEnv 2 Divide bar 8 Draw by scale 6 Expand 9 Fixed PT avg. 9 Font size 6,7 Histogram Editor 4,5 Horizontal Avg. 9 Incr. bar 8 Input Files 3.PDF files 3.CDF files 3.IDF files 3.SMP files 3 Interchanging variables 18 Independent 15 Known dep. 14 Max see Operations Merge bar 8 Midpoint Alg. 15 Min see Operations Monotonic variables 16 Operands 4 Operations 16 Combine 17 Max 16 Min 16 Parsing 16 Relational 17 Parsing see Operations 2 PDF mode 4,8 Pearson Correlation 13 Probability Distribution Function (PDF) 3 Prob. Interval Matrix 15 Prob. Matrix 14,15 Relational see Operations Simplex Method 19 Three or more 12 Transportation 19 Unknown dep. 13 Vertical Avg. 9 22