## The Tsshannoneffective Program - Calculate the Effective Shannon Probability

tsshannoneffective [-c] [-e] [-v] avg rms number

### DESCRIPTION

Tsshannoneffective is for calculating the effective Shannon probability, given the average, root mean square, data set size, and data set duration, of the normalized increments of a time series.

### DATA SET SIZE CONSIDERATIONS

This program addresses the question "is there reasonable evidence to justify investment in an equity based on data set size?"

The Shannon probability of a time series is the likelihood that the value of the time series will increase in the next time interval. The Shannon probability is measured using the average, avg, and root mean square, rms, of the normalized increments of the time series. Using the rms to compute the Shannon probability, P:

```
rms + 1
P = ------- ....................................(1.1)
2

```

However, there is an error associated with the measurement of rms do to the size of the data set, N, (ie., the number of records in the time series,) used in the calculation of rms. The confidence level, c, is the likelihood that this error is less than some error level, e.

Over the many time intervals represented in the time series, the error will be greater than the error level, e, (1 - c) * 100 percent of the time-requiring that the Shannon probability, P, be reduced by a factor of c to accommodate the measurement error:

```
rms - e + 1
Pc = ----------- ...............................(1.2)
2

```

where the error level, e, and the confidence level, c, are calculated using statistical estimates, and the product P times c is the effective Shannon probability that should be used in the calculation of optimal wagering strategies.

The error, e, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigma, is:

```
e
esigma = --- sqrt (2N) .........................(1.3)
rms

```

where N is the data set size = number of records. From this, the confidence level can be calculated from the cumulative sum, (ie., integration) of the normal distribution, ie.:

c esigma

50

0.67

68.27

1.00

80

1.28

90

1.64

95

1.96

95.45

2.00

99

2.58

99.73

3.00

Note that the equation:

```
rms - e + 1
Pc = ----------- ...............................(1.4)
2

```

will require an iterated solution since the cumulative normal distribution is transcendental. For convenience, let F(esigma) be the function that given esigma, returns c, (ie., performs the table operation, above,) then:

```
rms - e + 1
P * F(esigma) = -----------
2

rms * esigma
rms - ------------ + 1
sqrt (2N)
= ---------------------- .........(1.5)
2

```

Then:

```
rms * esigma
rms - ------------ + 1
rms + 1                      sqrt (2N)
------- * F(esigma) = ---------------------- ...(1.6)
2                            2

```

or:

```
rms * esigma
(rms + 1) * F(esigma) = rms - ------------ + 1 .(1.7)
sqrt (2N)

```

Letting a decision variable, decision, be the iteration error created by this equation not being balanced:

```
rms * esigma
decision = rms - ------------ + 1
sqrt (2N)

- (rms + 1) * F(esigma) ............(1.8)

```

which can be iterated to find F(esigma), which is the confidence level, c.

Note that from the equation:

```
rms - e + 1
Pc = -----------
2

```

and solving for rms - e, the effective value of rms compensated for accuracy of measurement by statistical estimation:

```
rms - e = (2 * P * c) - 1 ......................(1.9)

```

and substituting into the equation:

```
rms + 1
P = -------
2

```
```
rms - e = ((rms + 1) * c) - 1 .................(1.10)

```

and defining the effective value of rms as rmseff:

```
rmseff = rms - e ..............................(1.11)

```

It can be seen that if optimality exists, ie., f = 2P - 1, or:

```
2
avg = rms  ....................................(1.12)

```

or:

```                 2
avgeff = rmseff ...............................(1.13)

```

As an example of this algorithm, if the Shannon probability, P, is 0.51, corresponding to an rms of 0.02, then the confidence level, c, would be 0.996298, or the error level, e, would be 0.003776, for a data set size, N, of 100.

Likewise, if P is 0.6, corresponding to an rms of 0.2 then the confidence level, c, would be 0.941584, or the error level, e, would be 0.070100, for a data set size of 10.

Robustness is an issue in algorithms that, potentially, operate real time. The traditional means of implementation of statistical estimates is to use an integration process inside of a loop that calculates the cumulative of the normal distribution, controlled by, perhaps, a Newton Method approximation using the derivative of cumulative of the normal distribution, ie., the formula for the normal distribution:

```
2
1           - x   / 2
f(x) = ------------- * e           ............(1.14)
sqrt (2 * PI)

```

Numerical stability and convergence issues are an issue in such processes.

The Shannon probability of a time series is the likelihood that the value of the time series will increase in the next time interval. The Shannon probability is measured using the average, avg, and root mean square, rms, of the normalized increments of the time series. Using the avg to compute the Shannon probability, P:

```
sqrt (avg) + 1
P = -------------- ............................(1.15)
2

```

However, there is an error associated with the measurement of avg do to the size of the data set, N, (ie., the number of records in the time series,) used in the calculation of avg. The confidence level, c, is the likelihood that this error is less than some error level, e.

Over the many time intervals represented in the time series, the error will be greater than the error level, e, (1 - c) * 100 percent of the time-requiring that the Shannon probability, P, be reduced by a factor of c to accommodate the measurement error:

```
sqrt (avg - e) + 1
Pc = ------------------ .......................(1.16)
2

```

where the error level, e, and the confidence level, c, are calculated using statistical estimates, and the product P times c is the effective Shannon probability that should be used in the calculation of optimal wagering strategies.

The error, e, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigma, is:

```
e
esigma = --- sqrt (N) .........................(1.17)
rms

```

where N is the data set size = number of records. From this, the confidence level can be calculated from the cumulative sum, (ie., integration) of the normal distribution, ie.:

c esigma

50

0.67

68.27

1.00

80

1.28

90

1.64

95

1.96

95.45

2.00

99

2.58

99.73

3.00

Note that the equation:

```
sqrt (avg - e) + 1
Pc = ------------------ .......................(1.18)
2

```

will require an iterated solution since the cumulative normal distribution is transcendental. For convenience, let F(esigma) be the function that given esigma, returns c, (ie., performs the table operation, above,) then:

```
sqrt (avg - e) + 1
P * F(esigma) = ------------------
2

rms * esigma
sqrt [avg - ------------] + 1
sqrt (N)
= ----------------------------- .(1.19)
2

```

Then:

```
sqrt (avg)  + 1
--------------- * F(esigma) =
2

rms * esigma
sqrt [avg - ------------] + 1
sqrt (N)
----------------------------- .............(1.20)
2

```

or:

```
(sqrt (avg) + 1) * F(esigma) =

rms * esigma
sqrt [avg - ------------] + 1 .............(1.21)
sqrt (N)

```

Letting a decision variable, decision, be the iteration error created by this equation not being balanced:

```
rms * esigma
decision = sqrt [avg - ------------] + 1
sqrt (N)

- (sqrt (avg) + 1) * F(esigma) .....(1.22)

```

which can be iterated to find F(esigma), which is the confidence level, c.

There are two radicals that have to be protected from numerical floating point exceptions. The sqrt (avg) can be protected by requiring that avg >= 0, (and returning a confidence level of 0.5, or possibly zero, in this instance-a negative avg is not an interesting solution for the case at hand.) The other radical:

```
rms * esigma
sqrt [avg - ------------] .....................(1.23)
sqrt (N)

```

and substituting:

```
e
esigma = --- sqrt (N) .........................(1.24)
rms

```

which is:

```
e
rms * --- sqrt (N)
rms
sqrt [avg - ------------------] ...............(1.25)
sqrt (N)

```

and reducing:

```
sqrt [avg - e] ................................(1.26)

```

requiring that:

```
avg >= e ......................................(1.27)

```

Note that if e > avg, then Pc < 0.5, which is not an interesting solution for the case at hand. This would require:

```
avg
esigma <= --- sqrt (N) ........................(1.28)
rms

```

Obviously, the search algorithm must be prohibited from searching for a solution in this space. (ie., testing for a solution in this space.)

The solution is to limit the search of the confidence array to values that are equal to or less than:

```
avg
--- sqrt (N) ..................................(1.29)
rms

```

which can be accomplished by setting integer variable, top, usually set to sigma_limit - 1, to this value.

Note that from the equation:

```
sqrt (avg - e) + 1
Pc = ------------------
2

```

and solving for avg - e, the effective value of avg compensated for accuracy of measurement by statistical estimation:

```
2
avg - e = ((2 * P * c) - 1)  ..................(1.30)

```

and substituting into the equation:

```
sqrt (avg) + 1
P = --------------
2

```
```
2
avg - e = (((sqrt (avg) + 1) * c) - 1)  .......(1.31)

```

and defining the effective value of avg as avgeff:

```
avgeff = avg - e ..............................(1.32)

```

It can be seen that if optimality exists, ie., f = 2P - 1, or:

```
2
avg = rms  ....................................(1.33)

```

or:

```
rmseff = sqrt (avgeff) ........................(1.34)

```

As an example of this algorithm, if the Shannon probability, P, is 0.52, corresponding to an avg of 0.0016, and an rms of 0.04, then the confidence level, c, would be 0.987108, or the error level, e, would be 0.000893, for a data set size, N, of 10000.

Likewise, if P is 0.6, corresponding to an rms of 0.2, and an avg of 0.04, then the confidence level, c, would be 0.922759, or the error level, e, would be 0.028484, for a data set size of 100.

The Shannon probability of a time series is the likelihood that the value of the time series will increase in the next time interval. The Shannon probability is measured using the average, avg, and root mean square, rms, of the normalized increments of the time series. Using both the avg and the rms to compute the Shannon probability, P:

```
avg
--- + 1
rms
P = ------- ...................................(1.35)
2

```

However, there is an error associated with both the measurement of avg and rms do to the size of the data set, N, (ie., the number of records in the time series,) used in the calculation of avg and rms. The confidence level, c, is the likelihood that this error is less than some error level, e.

Over the many time intervals represented in the time series, the error will be greater than the error level, e, (1 - c) * 100 percent of the time-requiring that the Shannon probability, P, be reduced by a factor of c to accommodate the measurement error:

```
avg - ea
-------- + 1
rms + er
P * ca * cr = ------------ ....................(1.36)
2

```

where the error level, ea, and the confidence level, ca, are calculated using statistical estimates, for avg, and the error level, er, and the confidence level, cr, are calculated using statistical estimates for rms, and the product P * ca * cr is the effective Shannon probability that should be used in the calculation of optimal wagering strategies, (which is the product of the Shannon probability, P, times the superposition of the two confidence levels, ca, and cr, ie., P * ca * cr = Pc, eg., the assumption is made that the error in avg and the error in rms are independent.)

The error, er, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigmar, is:

```
er
esigmar = --- sqrt (2N) .......................(1.37)
rms

```

where N is the data set size = number of records. From this, the confidence level can be calculated from the cumulative sum, (ie., integration) of the normal distribution, ie.:

cr esigmar

50

0.67

68.27

1.00

80

1.28

90

1.64

95

1.96

95.45

2.00

99

2.58

99.73

3.00

Note that the equation:

```
avg
-------- + 1
rms + er
P * cr = ------------ .........................(1.38)
2

```

will require an iterated solution since the cumulative normal distribution is transcendental. For convenience, let F(esigmar) be the function that given esigmar, returns cr, (ie., performs the table operation, above,) then:

```
avg
-------- + 1
rms + er
P * F(esigmar) = ------------ =
2

avg
------------------- + 1
esigmar * rms
rms + -------------
sqrt (2N)
----------------------- ......(1.39)
2

```

Then:

```
avg
--- + 1
rms
------- * F(esigmar) =
2

avg
------------------- + 1
esigmar * rms
rms + -------------
sqrt (2N)
----------------------- ................(1.40)
2

```

or:

```
avg
(--- + 1) * F(esigmar) =
rms

avg
------------------- + 1 ................(1.41)
esigmar * rms
rms + -------------
sqrt (2N)

```

Letting a decision variable, decision, be the iteration error created by this equation not being balanced:

```
avg
decision =  ------------------- + 1
esigmar * rms
rms + -------------
sqrt (2N)

avg
- (--- + 1) * F(esigmar) ..........(1.42)
rms

```

which can be iterated to find F(esigmar), which is the confidence level, cr.

The error, ea, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigmaa, is:

```
ea
esigmaa = --- sqrt (N) ........................(1.43)
rms

```

where N is the data set size = number of records. From this, the confidence level can be calculated from the cumulative sum, (ie., integration) of the normal distribution, ie.:

ca esigmaa

50

0.67

68.27

1.00

80

1.28

90

1.64

95

1.96

95.45

2.00

99

2.58

99.73

3.00

Note that the equation:

```
avg - ea
-------- + 1
rms
P * ca = ------------ .........................(1.44)
2

```

will require an iterated solution since the cumulative normal distribution is transcendental. For convenience, let F(esigmaa) be the function that given esigmaa, returns ca, (ie., performs the table operation, above,) then:

```
avg - ea
-------- + 1
rms
P * F(esigmaa) = ------------ =
2

esigmaa * rms
avg - -------------
sqrt (N)
------------------- + 1
rms
----------------------- ......(1.45)
2

```

Then:

```
avg
--- + 1
rms
------- * F(esigmaa) =
2

esigmaa * rms
avg - -------------
sqrt (N)
------------------- + 1
rms
----------------------- ................(1.46)
2

```

or:

```
avg
(--- + 1) * F(esigmaa) =
rms

esigmaa * rms
avg - -------------
sqrt (N)
------------------- + 1 ................(1.47)
rms

```

Letting a decision variable, decision, be the iteration error created by this equation not being balanced:

```
esigmaa * rms
avg - -------------
sqrt (N)
decision = ------------------- + 1
rms

avg
- (--- + 1) * F(esigmaa) ..................(1.48)
rms

```

which can be iterated to find F(esigmaa), which is the confidence level, ca.

Note that from the equation:

```
avg
-------- + 1
rms + er
P * cr = ------------
2

```

and solving for rms + er, the effective value of rms compensated for accuracy of measurement by statistical estimation:

```
avg
rms + er = ---------------- ...................(1.49)
(2 * P * cr) - 1

```

and substituting into the equation:

```
avg
--- + 1
rms
P = -------
2

```
```
avg
rms + er = -------------------- ...............(1.50)
avg
((--- + 1) * cr) - 1
rms

```

and defining the effective value of avg as rmseff:

```
rmseff = rms +/- er ...........................(1.51)

```

Note that from the equation:

```
avg - ea
-------- + 1
rms
P * ca = ------------
2

```

and solving for avg - ea, the effective value of avg compensated for accuracy of measurement by statistical estimation:

```
avg - ea = ((2 * P * ca) - 1) * rms ...........(1.52)

```

and substituting into the equation:

```
avg
--- + 1
rms
P = -------
2

```
```
avg
avg - ea = (((--- + 1) * ca) - 1) * rms .......(1.53)
rms

```

and defining the effective value of avg as avgeff:

```
avgeff = avg - ea .............................(1.54)

```

As an example of this algorithm, if the Shannon probability, P, is 0.51, corresponding to an rms of 0.02, then the confidence level, c, would be 0.983847, or the error level in avg, ea, would be 0.000306, and the error level in rms, er, would be 0.001254, for a data set size, N, of 20000.

Likewise, if P is 0.6, corresponding to an rms of 0.2 then the confidence level, c, would be 0.947154, or the error level in avg, ea, would be 0.010750, and the error level in rms, er, would be 0.010644, for a data set size of 10.

As a final discussion to this section, consider the time series for an equity. Suppose that the data set size is finite, and avg and rms have both been measured, and have been found to both be positive. The question that needs to be resolved concerns the confidence, not only in these measurements, but the actual process that produced the time series. For example, suppose, although there was no knowledge of the fact, that the time series was actually produced by a Brownian motion fractal mechanism, with a Shannon probability of exactly 0.5. We would expect a "growth" phenomena for extended time intervals [Sch91, pp. 152], in the time series, (in point of fact, we would expect the cumulative distribution of the length of such intervals to be proportional to erf (1 / sqrt (t)).) Note that, inadvertently, such a time series would potentially justify investment. What the methodology outlined in this section does is to preclude such scenarios by effectively lowering the Shannon probability to accommodate such issues. In such scenarios, the lowered Shannon probability will cause data sets with larger sizes to be "favored," unless the avg and rms of a smaller data set size are "strong" enough in relation to the Shannon probabilities of the other equities in the market. Note that if the data set sizes of all equities in the market are small, none will be favored, since they would all be lowered by the same amount, (if they were all statistically similar.)

To reiterate, in the equation avg = rms * (2P - 1), the Shannon probability, P, can be compensated by the size of the data set, ie., Peff, and used in the equation avgeff = rms * (2Peff - 1), where rms is the measured value of the root mean square of the normalized increments, and avgeff is the effective, or compensated value, of the average of the normalized increments.

### DATA SET DURATION CONSIDERATIONS

An additional accuracy issue, besides data set size, is the time interval over which the data was obtained. There is some possibility that the data set was taken during an extended run length, either negative or positive, and the Shannon probability will have to be compensated to accommodate this measurement error. The chances that a run length will exceed time, t, is:

```
1 - erf (1 / sqrt (t)) ........................(1.55)

```

or the Shannon probability, P, will have to be compensated by a factor of:

```
erf (1 / sqrt (t)) ............................(1.56)

```

giving a compensated Shannon probability, Pcomp:

```
Pcomp = Peff * (1 - erf (1 / sqrt (t)))........(1.57)

```

Fortunately, since confidence levels are calculated from the normal probability function, the same lookup table used for confidence calculations (ie., the cumulative of a normal distribution,) can be used to calculate the associated error function.

To use the value of the normal probability function to calculate the error function, erf (N), proceed as follows; since erf (X / sqrt (2)) represents the error function associated with the normal curve:

1. X = N * sqrt (2).

2. Lookup the value of X in the normal probability function.

3. Subtract 0.5 from this value.

4. And, multiply by 2.

or:

```
erf (N) = 2 * (normal (t * sqrt (2)) - 0.5) ...(1.58)

```

### OPTIONS

avg
Average of the normalized increments of the time series.

rms
Root mean square of the normalized increments of the time series.

number
Number of records used to calculate avg and rms.

-c
Compensate the Shannon probability for run length duration.

-e
Print only erf (1 / sqrt (number)), 1 - erf (1 / sqrt (number)).

-v
Print the version and copyright banner of this program.

### WARNINGS

There is little or no provision for handling numerical exceptions.

tsinvest(1), tsinvestsim(1), tsshannoneffective(1)

### DIAGNOSTICS

Error messages for incompatible arguments, failure to allocate memory, inaccessible files, and opening and closing files.

### AUTHORS

A license is hereby granted to reproduce this software source code and to create executable versions from this source code for personal, non-commercial use. The copyright notice included with the software must be maintained in all copies produced.

THIS PROGRAM IS PROVIDED "AS IS". THE AUTHOR PROVIDES NO WARRANTIES WHATSOEVER, EXPRESSED OR IMPLIED, INCLUDING WARRANTIES OF MERCHANTABILITY, TITLE, OR FITNESS FOR ANY PARTICULAR PURPOSE. THE AUTHOR DOES NOT WARRANT THAT USE OF THIS PROGRAM DOES NOT INFRINGE THE INTELLECTUAL PROPERTY RIGHTS OF ANY THIRD PARTY IN ANY COUNTRY.

So there.

john@email.johncon.com

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