PPT NOTES, V.10

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Appl. Math Seminar, 223 Weber, Thursday, Sept.14, 2017

Title: Theodorus Spiral, Vector Algebra andSystem Monitoring
Speaker: T. Jay Bai, Ph.D. 970-495-9716, tjbmdsm@yahoo.com

Abstract:

“Theodorus Spiral” can be expressed as a 17*17diagonal matrix, and then further as a 17-vector. Vectors are variableswith both magnitude and directions. They are extension of scalars upon whichthe Group of multiplication apply. So they can be used for system analysis.Vector length expresses the sum of the components, while the directions carrythe composition information of the systems. Rotations of the vectors in17-space are considered as system successional trends, and are expressed byratio of present and previous cosine values. Our trend analysis model has beentested in financial system by investment management and checked by government.Furthermore, a time chain (Bai-Jameson Chain) based on trends has beendeveloped for system prediction and monitoring.

Essential phrases:

The Vector_(𝑖), 𝑖=1,2,…𝑚， are the extension of scalars, and the scalars are thespecial cases of  𝑉𝑒𝑐𝑡𝑜𝑟𝑠_(i) , 𝑖=1.

The 17-direction of 17-vector in 17-space expresses the composition of the 17-systems, and the composition is essential for systems.

Host: Prof. James Liu

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My name is TugjayzhabJay Bai. I am a quantitative Ecologist, and specialize in ecosystem-monitoring.To discover the degradation trends of grasslands in Inner Mongolia, China,1984, I have started using inverse of a diagonal matrices to express the trendsand using hyper cosine value as correlation coefficient to classify grasslands.Based on collected information and data, I predicted that the increasing trendsof grassland degradation, and even dust storm increasing trend around Beijing,1984. However, people did not like the information to be released, it wasclassified as “SECRET”.

But I believe the trendis a true reality, and the research is important. As our world is made ofsystems, instead of variables, it is important that we can monitor our naturalsystems. I continue working on Multivariate data analysis methods for aproposed Grassland Monitoring Net.  WhenI was working for LCTA, CSU, I have developed a system monitoring model, socalled “Multi-Dimensional Sphere Model”, now I think it should be called VECTORALGEBRA, compare with the scalar algebra and mat algebra （linear algebra).

You all have learnedy=f(x), expressing the relation among the variables. In this presentation, Iwill introduce  you a simple, easy, yetpowerful way to discover the relation among the systems:

𝑖=1, y=f(x) çè𝑌_(𝑖)=𝑓(𝑋_(𝑖)), 𝑖=1,2,3,…17

And I expecting youtelling me: this method is mathematically sound, and my English isunderstandable.

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Lets start with a 2500 year old Theodorus Spiral

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According to Wikipedia ^[1], about 2,500 years ago, in order to prove that the square root of non-square numbers are irrational numbers, Theodorus of Cyrene (5th-century BC) proposed the"Theodorus Spiral".

A 2-dimensional Theodorus Spiral is demonstrated here,

The outer wheel, ∆=1

Diameter, ∆=√(𝑖+1)−√𝑖

Rotation, 𝜃=𝑎𝑟𝑐𝑡𝑎(1⁄√𝑖),

𝑖=1,2,…17

It stopped on 17 in 2-space

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Next topic, DiagonalMatrix

Now, the sequential 17 ones(unit vectors) that are orthogonal to each other, can be expressed as a Matrix

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Since the seventeen ’s=1 are mutuallyperpendicular to each other in 17-space, they form a 17*17 Diagonal Matrix. Wenamed it "Theodorus-Matrix", or "Th-Matrix". The diagonalelements of this 17*17 "Th-Matrix" are equal to 1, and thenon-diagonal elements are zeros:

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Th-Matrix is the "Identity Matrix", which is a matrix of multiplicative identity, and has afew special properties:
1) It can be multiplied by the left or right by a same dimensional matrix, but remains the same:
I*A = A*I = A

2) If the product of two matrices A and B is an Identity Matrix, these two matrices are mutuallyinverse to each other:
If A*B = B*A = I, then
A = B^-1, and B = A^-1

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If the diagonal elementsof the "Th-matrix" are replaced by non-zero real numbers, then itbecomes a normal diagonal matrix and a normal diagonal matrix is aMultiplicative Group. Then the "Th-matrix" has some practicalapplications. For example, if we replaced the diagonal elements with 17 dailyclosing prices of 17 funds, such as: DIA, FBALX,… SPY, then we formed a Th-MarketDiagonal Matrix that expresses the 17-fund market.

DIA is the Do JonesIndustrial Average, SPY is the Standard & Poor 500 ETF

For a rangeland, thecomponents would be plant species, Chinese grassland, would be STIPA, LEYMUS,SALSOLA,…

For Fort Carson LandCondition Trend Analysis, they would be: Bromus, Bouteloua, Yucca, Pinus,etc.

For Yellow Stonevegetation recovery project, the components would be Tree, Shrub, Grasses.

For anti-random test EE,they would be for example, 6-bit numbers.

Here we are using stockmarket variables, as these are the huge data set updated timely, accurate,accessible by anyone, and most important charge free.

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The stocks are not for free, so they allnon-zero, positive number,

Their reciprocals make the inverse of  Th-Market Mat.

Th-Market Mat can be r-multiplied and l-multiplied,and the product is another 17-vector. It means that the multiplication isclose;

Has inverse,

Has unit

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The Th-Market Matrices is a Group of multiplication.

We can do lots of calculation (basic arithmetic) on diagonalmatrix just like we do on real numbers.

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First, we calculatemarket total, expressed by SSS.

You may call SSS asEuclidean Norm, but we name it as The Th-sum.

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STANDARDIZATION

While the Sum is available, we divide each andevery elements by total to calculate the percentage. This special percentage,Theodorus Percentage, we name it “Importance Value”. As this model orientedfrom vegetation science. The dominance, the importance value, is an essentialfeature for vegetation.

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For system monitoring, we introduce time index: k, k-1, k-2

Then we define the quotients of present (k) over previous (k-1) IV as the SystemSuccessional Trends of the market, and each/every fund ^[2]:

Trend_(i,k)= IV_(i,k)/IV_{(i, k-1) ,} i=1,2,…17

The expectation of the trend values are 1:

Trend_(i,k)=IV_(i,k)/IV_{(i, k-1)}  =1, where i=1,2,...17, means that IV  is no change. If

Trend_(i,k)=IV_(i,k)/IV_{(i, k-1)}  >1, where i=1,2,...17, means that the IV of given fund is increasing, but

Trend_(i,k)=IV_(i,k)/IV_{(i, k-1)}  <1, where i=1,2,...17, means that the IV of given fund is decreasing.

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The expectation of thetrends are one, which means, we suppose the COMPOSITION of a system hasinertia. Composition of a system tend to remain the same, even when the Sumchanges. Thus, Composition is essential for the systems.

If we use threedimension of tree, shrubs, grass to describe vegetation:, then 3-2-1: 3-tree,2-shrub, 1-grass is totally different vegetation from 1-2-3: 1-tree, 2-shrubs,3-grasses, even the total is 6 in both cases. And on the same time 30-20-10 isthe same as 3-2-1.

A one inch photo is thesame as an eight inches photo. Scalar multiplication will not essentiallychange systems.

This is very important,and essential for vegetation, and other natural resource sharing system. So, weneed a new kind of variables to describe both quantity and quality of a system.This kind of variable DOES EXIST.

On the other hand,Vectors are variables with magnitude and directions. And their directions areessential for vectors. Co-liners have the same projection on the unithypersphere.

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Vectors are the variables with both Magnitudeand Directions. They can be used to express both quantity and quality ofsystems.

Magnitude of vectors express the sum of thecomponents of a system, and the directions (pl) express the composition of thesystems.

Even though everyone accept the definition ofthe vectors, not many people know what are the directions of a 17-vector in17-space.

In MDSM, directions of a 17-vector is expressedby 17-cosine values:

𝜃_(𝑖)=𝐴𝑟𝑐𝑐𝑜𝑠_(𝑖) , i=1,2,…17, and

The directions of a vector meaning the Distributionof the total in the systems. Or say:

The directions of a vectors show the COMPOSITIONof the systems.

All the powered cosine values equals one: ∑(𝑐𝑜𝑠_(𝑖))² =1 , 𝑖=1,2,…17

The 17-vector_(𝑖), i=1,2,3,…17, is an extension of scalar, while scalar is aspecial case of vector_(𝑖), i=1.

Certainly, the vectors are NOT n*1 matrix.

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AX=B → X=B/A, C=BX,…

Other than save space,with both quality and quantity, vectors are qualified as a system monitoringtools, as it is a Group of multiplication, provided a component wisemultiplication.

All of us uses worksheetand do the vector multiplication all the time, now. People may take it asgranted, but that is not true. In science research, nothing is granted. Theremust be someone who worked out, put in publication or registered for copyright.WHO is the first one to define component wise Vector multiplication, andinverse? If there is no citation stating otherwise, then I would say, the CSUresearch associate, Dr. T. Jay Bai, is the first one to give definition ofvector component wise multiplication, inverse, and division, (registered forcopyright and published paper on 1995 and 1997, respectively.) That makesvectors a Group of Multiplication, and makes system transition available, andmakes System Monitoring possible.

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Here is the citation.

The definition of vectors multiplication was submitted on 1995, but published on 1997.

Before, there were no vector multiplication, nor division. If you multiply two vectors, thecomputer will be responded a scalar (dot product), or a matrix (cross product).

Till 1995, Borland, QUOTRO PRO version 5, supply the vector multiplication. The product of  two 17-vector is another 17-vector: components of product are the products of components.That makes vectors a closed set.

Ecological Modelling 97 (1997)75-86

Received 25 August 1995; accepted 15 August 1996

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It was also registeredwith copyright office.

Stock Market Monitor 52,copyrighted on May 24, 1995

The computer programused vectors division (trends), multiplication (Prediction).

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Thus, the 𝑉𝑒𝑐𝑡or_(𝑖), 𝑖=1,2,…𝑚， are the extension of scalars,and the scalars are the special cases for 𝑉𝑒𝑐𝑡𝑜𝑟𝑠_(𝑖), 𝑖=1.

Number set is extended from scalar to m-Vectors.

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Extend our discussion from 17*17 Matrices to17-vectors:

STANDARDIZATION: Each and every componentdivided by SSS, vector length. This projects a 17-vector in 17-space on to theunit hypersphere, and expresses the directions of 17-market vector with theircosine values.

The standardization OMITED MAGNITUDE, but leftsystem composition information: All vectors magnitudes equal one, but emphasizethe directions, which expressed by17- cos’, representing the systemscomposition.

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Back to continue our discussion on systemmonitoring.

For system monitoring we bring in time indices:K, K-1, K-2

Instead of using IV, now we use cosine toexpress the state of vectors:

Then the system successional trends are defined aspresent cosine over the previous cosine values:

Trend _(i,k) = cos _(i，k)/cos _(i,k-1)

The expectation of the trend values are 1:

Trend_(i,k)=COS_(i,k)/COS_(i,k-1)  =1, where i=1,2,...17, means that there is no change

Trend_(i,k)=COS_(i,k)/COS_(i,k-1)  >1, where i=1,2,...17, means that the given fund isincreasing, but

Trend_(i,k)=COS_(i,k)/COS_(i,k-1)  <1, where i=1,2,...17, means that the given fund isdecreasing.

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Further, we define the Trends of Trends as second degree Trends, TT.

TT_(k+1)=T_(k+1)/ T_(k)

=[𝐶𝑂𝑆_(𝑘+1)/𝐶𝑂𝑆_𝑘]/[𝐶𝑂𝑆_𝑘/𝐶𝑂𝑆_(𝑘−1)]

= [𝐶_(𝑘+1)*𝐶𝑂𝑆_(𝑘−1)]/𝐶𝑂𝑆_𝑘^2],

The TT compares the two ending points with the middle point, to show the turning points inthe time series.

The expectation values of the TT are 1:

TT_(k+1) = [𝐶𝑂𝑆_(𝑘+1)*𝐶𝑂𝑆_(𝑘−1)]/𝐶𝑂𝑆_𝑘^2] =1, shows that there is no turning; while,

TT_(k+1) = [𝐶𝑂𝑆_(𝑘+1)*𝐶𝑂𝑆_(𝑘−1)]/𝐶𝑂𝑆_𝑘^2] > 1, means that the product of the two ending pointsis higher than the squared middle point, so the turning exists, and itis concave; but, if

TT_(k+1) = [𝐶𝑂𝑆_(𝑘+1)*𝐶𝑂𝑆_(𝑘−1)]/𝐶𝑂𝑆^2] < 1, means that the product of the two ending pointsis less than the squared middle point, so the turning exists, but itis convex.

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Combining the three trends: up, even, or down, with three turns (TT): concave, straight, orconvex, these nine states describes each of the 17 funds and market more clearly than usingscalars before. Then the 17 funds can be put in order by their Trend Values and/orSecond Degree Trend Values, to help investors make decisions, as we suppose that the marketsystems have some inertia in their neighborhood.

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So far, we have been logically reasoning the system monitoring.

Before go further, let me give a summary:

Our world is made of systems, a single variable has to be put in a system to make meaning.For example a stock, it looks like a random variable, jumping around, but after putting in market, theImportance Value would be a constant.

If a 17-fund market system be expressed by a 17-vector, the vector length expresses thesum of the systems, but the 17-direction of the vectors in 17-space expresses the compositionof the 17-system. The composition is essential for systems. Scalar multiplicationwould not change the systems, only the vector multiplication can change the systems.

For example, A=3-2-1 is different from B=1-2-3, but same as A10=30-20-10. As the two are co-liners. A market expressed by #, dollar, yin, looks different, but it is the same market.

Vector is also a multiplication Group. We can do very basic arithmetic.

So far,

IT IS SIMPLE, IS NOT IT?

IT IS EASY, IS NOT IT?

Now we are going to cases of applications, that is my favor, I will tell you, it is not only simpleand easy, it is also powerful, can solve many problems that scalar, matrices, statistics couldnot do before.

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IN LAST THIRTY YEARS, I HAVECONDUCTED SOME OF THE APPLICATIONS

1 Vegetation Classification, 1981 using cos as similarity coefficient

Shalawplaw and Leymus recovering, 1983, using diagonal mat as state transition matrices

Grassland degradation in China 84, predict the dust storm increasing trend Beijing area:  y=-38+1.1x, it actually a regression analysis.

Land Condition Trend Analysis, @ Fort Carson; land usage map using trend index, |trends|

Investment tests: ESA2001

Six years IRA, checked by authorities.

Anti-random test

Yellow stone model

Predict results @ Italy from CSU

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Let’s start with the original one

Instead of using total production, that affected by precipitation, I used component comparing. Ifound that good grasses were going down, but the bad grasses were going up.So my conclusion was grasslands were degrading. To attract people’s attention, I combined with weatherprediction, say the strong wind increasing trend exist, even Beijing area.

This is the report of grassland degradation. Submitted on August 1984, stamped with “secret”.

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The trend of dust storm at Beijing area is increasing:

Y=-38+1.1x

Later the trend was proved to be true, I feel very sorry. I would rather my prediction was wrong.

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Ten years later. LCTA (Land Condition Trend Analysis)program: land usage map of Fort Carson.

MDSM was applied to Land Condition Trend Analysis project at Fort Carson, Colorado USA.The map based on the trend index was matching the range manager’ experience. The report was presented to theESA, Snowbird UT, 1995 and is published in Ecological Modelling 1997.

The paper gave the definition of component wise production, as mentioned above.

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My proposal did not get funded. I left LCTA, but continued my research without funding andwithout salary. But the ecological data are expensive, and not very accurate, not available topublic. So I need to find out another data resource to polish my research.

I found the stock data, they are: accurate, available to every one, updated timely for no cost.

Here is the most attractive case of investment in stock market.

Steps of Application of Trend Analysis onto financial investment.

Define a 20 Funds market.

Vector by column: Variables code, name; k-2, k-1, k, the time unit was week. Average of k-2, k-1, k, Trends(k-1, Trends k, and TT.

Reordered by trends k values.

Distribute your invest into several stocks, and keeps more than half of your holdings above half,then you can gain over the average market.

Change fund of COLGX to STMAX. Keeps the better ones, but give away the worst.

For more information, please visit my current blog, column of CLUB: http://blog.sciencenet.cn/home.php?mod=space&uid=333331&do=blog&classid=171794&view=me&from=space

There are my lecture notes, and MDSM club meeting minutes.

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This is the second year’ report 1999.  Totally 18 months experiment. The second part started 99-01-29, ended 00-01-28. The return during 12 months were 32%, on the same period, market increased 9.82%. Thereport paper was published at University of Inner Mongolian, China, slides were presented to theESA 2001.

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The longest and the most excited case is here:

A personal IRA account managed by MDSM: Jan. 1998-April 2004

MDSM Gain: 109%, vs.

DIA, 29%

NASDAQ 19%

S&P500 13%

Interestingly, the results from MDSM ‘s market validation since mid 1998 among the top ½ of 1percent of all 8126 fund managers in the United States.

Warren Stobbe

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This flyer draw authority’ attention.

Here is a summon letter from the office of Investment Company/Investment Adviser Regulation, Securities and Exchange Commission Central Regional Office

They want to see my records supporting the above graphics.

Before our meeting, it looks like they found the record data themselves already.

I brought all my six years statementswith me. They accepted, and told me that I can use that graphic to demo my research, but before I get the license, I can not do the business.

http://blog.sciencenet.cn/blog-333331-982092.html

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But I know I am not good at business, only suit for research.  I also know my research is veryimportant. It has discovered something nobody has ever done before, to monitor our ecosystems, where and how fast they are going.

Next I explain how this work for a proposed “grassland monitoring net”, or any other system monitoring programs, multi-variate time series.

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TRENDS ARE USED AS THE TRANSITION VECTOR OF THE SYSTEMS

Predict for next step, then collect data to verify the prediction;

As both Prediction and Collected Data have errors (prediction error and sampling error), not thetrue value, we combine the prediction and the collected data to generate weighing average asexpectations of the true values. It would be the closest to true values with less error.

Then, continue using Expectations to calculate Cosines, then calculate Trends, topredict the next-next step, …and so on.

The two weighing factors 𝛼 and 𝛽 can be calculated from the variances of the P and D. Biggervariance has a less weight. If we don’t know exactvalues, then let them equal, half and half, easy and simple.

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After 𝐸_𝑘 replaced by (𝐷_𝑘 + 𝑃_𝑘 ), and then 𝑃_𝑘 replaced by (𝐸_(𝑘−1)* 𝑇_(𝑘−1)),…then we will have a data chain (time series) from time 0 to time k, used to predict 𝑃_(𝑘+1), the system state on timek+1.

This chain named as “Bai-Jameson Chain”. Dr. Donald Jamesonwas a professor at Natural Resource College, Range Science Department, later Forest Department. He was in my committee member, and taught me NR 675, Environmental Monitoring and Adaptive Management.

After my graduation, while I was working for Fort Carson, he continued helping me byproofreading my papers, giving me suggestions and advises. He actually somehow helped medeveloped the MDSM. However, he might not have realized that before he passed away.

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As 𝑃_(𝑘+1) expressed by a series T and D, from time k back to time 0, and T can be expressed by D, T=f(d), thus, prediction used ALL the history data, but ONLY the history data.

That makes the prediction available to every body, even high school students with noexperiences.

Cf: Some other time series using possibilities requires experience.

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Here is a sample that I predicted an experimental result in Italy.

A biologist named Macro from Italy supplied the original data of five species, 14 seriesobservations to ECOLOG-L, 1998, and asked for help:

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I put his original data into my DOS executable SMM-52, and predicted a series.This is the 15^th, used all 14 steps before:

The first column is the variable names， such as plant species, A, B, C, D, and E; V-SUM (Vector Length), Cosine, and SMC (System Monitoring Coefficient).

P: projection values of the given year based on the previous information.

D: Observation Data. These are the values that the program read from the input data that he supplied.

E: Expectation. It is the weighted averages from projections and observations, E=(D+P)/(alpha+beta).

I.V.: Importance Values. It is the importance values, or relative composition, of the species.

R: Projection Error. (R=E-P. Please notice that, the error definition is the difference between the projections and expectations.)

T: Multivariate Instantaneous Trend. T=IV_(k)/IV_(k-1). This model analyzes the composition change of the vegetation. It considers the composition change is the essential change of the vegetation, or any community.

P_(k+1): Vegetation Projection for the next year. (k is the index of the year.)

V-SUM: vector sum, or Vector Lengths, describe the general situation of the vegetation.

Cos: Cosine values between two vectors describing the correlation between the two vectors.(The Cosine values under column D express the correlation between the observations:

cosine<D_(k), D_(k-1)>, while the values under column E express the correlation between theobservation and expectation: cosine<D,E>.)

SMC: System Monitoring Coefficient,SMC=Cos<E_(k),D_(k)>/Cos<D_(k), D_(k-1)>.When the SMC value is greater than or close to one, we consider that the model is working forsystem monitoring and it's projections were over fitting or correct fitting the observations.

In your original email, you asked that:" What would the next sequence of number will be?"  As a temporal dynamic analysis model, the MDSM projected the values for the 15th year'svegetation. The values for the plant species A, B, C, D, and E, are:

597.59, 255.02, 1133.47, 1408.26, and 410.30, respectively.

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And predicted 16^th results after M provided 15^th observation

The projection for 16th observation was:

460, 232, 1148, 1417, and 350.

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Comparison of predicted and collected 16^th observation.

http://blog.sciencenet.cn/blog-333331-1072699.html

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Above is a brief summary of my research in last 35 years.

NOW, before you ask me questions, let me ask you:

IT IS POWERFUL, IS NOT IT? MY ENGLISH IS UNDERSTANDABLE, IS NOT IT?

This field used to be called “Multivariate Analysis”. A famous Prof. is HughGauch, @ Cornell University. He published his book: Multivariate Analysis in Community Ecology, 1982. Attention: community ecology. Follow his steps, I started with Cosine value asCorrelation Coefficient for Vegetation classification, using diagonal matrix for statetransition, then come to Group.  What doVECTORS mean? It means that variables are independent. All the variables that formed a community/systems areindependent because of resource sharing/competing. Competing makes the systemsless efficient, less productive, but it also makes systems sustainable. United/Harmony butIndependent (和而不同), that might be the universal rule.  

It is said: the world are numbers. I interpret it as “the world are vectors, instead ofscalars, as the world are systems, instead of variables.

We are facing a big change: from Y=f(X) to 𝑌_(𝑖) = f(𝑋_(𝑖)), 𝑖=1,2,…𝑚. Even the leap from scalar to vector is big, but it is easy, simple, isn’t it?

I will answer your questions, now.