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added code to make a list of CI for an alpha for a range of numbers of observations.

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[SVN r39487]
This commit is contained in:
Paul A. Bristow
2007-09-23 17:29:17 +00:00
parent 80354ab802
commit d262d6d4c3
1 changed files with 183 additions and 64 deletions
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247
example/chi_square_std_dev_test.cpp
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@@ -7,14 +7,18 @@
// or copy at http://www.boost.org/LICENSE_1_0.txt)
#include <iostream>
using std::cout; using std::endl;
using std::left; using std::fixed; using std::right; using std::scientific;
#include <iomanip>
using std::setw;
using std::setprecision;
#include <boost/math/distributions/chi_squared.hpp>
void confidence_limits_on_std_deviation(
double Sd, // Sample Standard Deviation
unsigned N) // Sample size
{
//
// Calculate confidence intervals for the standard deviation.
// For example if we set the confidence limit to
// 0.95, we know that if we repeat the sampling
@@ -27,8 +31,10 @@ void confidence_limits_on_std_deviation(
// contains the true standard deviation or it does not.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm
using namespace std;
using namespace boost::math;
// using namespace boost::math;
using boost::math::chi_squared;
using boost::math::quantile;
using boost::math::complement;
// Print out general info:
cout <<
@@ -40,11 +46,9 @@ void confidence_limits_on_std_deviation(
cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";
//
// Define a table of significance/risk levels:
//
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
//
// Start by declaring the distribution we'll need:
//
chi_squared dist(N - 1);
//
// Print table header:
@@ -56,7 +60,6 @@ void confidence_limits_on_std_deviation(
"_____________________________________________\n";
//
// Now print out the data for the table rows.
//
for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
{
// Confidence value:
@@ -69,7 +72,57 @@ void confidence_limits_on_std_deviation(
cout << fixed << setprecision(5) << setw(15) << right << upper_limit << endl;
}
cout << endl;
}
} // void confidence_limits_on_std_deviation
void confidence_limits_on_std_deviation_alpha(
double Sd, // Sample Standard Deviation
double alpha // confidence
)
{ // Calculate confidence intervals for the standard deviation.
// for the alpha parameter, for a range number of observations,
// from a mere 2 up to a million.
// O. L. Davies, Statistical Methods in Research and Production, ISBN 0 05 002437 X,
// 4.33 Page 68, Table H, pp 452 459.
// using namespace std;
// using namespace boost::math;
using boost::math::chi_squared;
using boost::math::quantile;
using boost::math::complement;
// Define a table of numbers of observations:
unsigned int obs[] = {2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 40 , 50, 60, 100, 120, 1000, 10000, 50000, 100000, 1000000};
cout << // Print out heading:
"________________________________________________\n"
"2-Sided Confidence Limits For Standard Deviation\n"
"________________________________________________\n\n";
cout << setprecision(7);
cout << setw(40) << left << "Confidence level (two-sided) " << "= " << alpha << "\n";
cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";
cout << "\n\n" // Print table header:
"_____________________________________________\n"
"Observations Lower Upper\n"
" Limit Limit\n"
"_____________________________________________\n";
for(unsigned i = 0; i < sizeof(obs)/sizeof(obs[0]); ++i)
{
unsigned int N = obs[i]; // Observations
// Start by declaring the distribution with the appropriate :
chi_squared dist(N - 1);
// Now print out the data for the table row.
cout << fixed << setprecision(3) << setw(10) << right << N;
// Calculate limits: (alpha /2 because it is a two-sided (upper and lower limit) test.
double lower_limit = sqrt((N - 1) * Sd * Sd / quantile(complement(dist, alpha / 2)));
double upper_limit = sqrt((N - 1) * Sd * Sd / quantile(dist, alpha / 2));
// Print Limits:
cout << fixed << setprecision(4) << setw(15) << right << lower_limit;
cout << fixed << setprecision(4) << setw(15) << right << upper_limit << endl;
}
cout << endl;
}// void confidence_limits_on_std_deviation_alpha
void chi_squared_test(
double Sd, // Sample std deviation
@@ -85,8 +138,11 @@ void chi_squared_test(
// that any difference is not down to chance.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm
//
using namespace std;
using namespace boost::math;
// using namespace boost::math;
using boost::math::chi_squared;
using boost::math::quantile;
using boost::math::complement;
using boost::math::cdf;
// Print header:
cout <<
@@ -145,21 +201,23 @@ void chi_squared_test(
else
cout << "REJECTED\n";
cout << endl << endl;
}
} // void chi_squared_test
void chi_squared_sample_sized(
double diff, // difference from variance to detect
double variance) // true variance
{
using namespace std;
using namespace boost::math;
// using boost::math;
using boost::math::chi_squared;
using boost::math::quantile;
using boost::math::complement;
using boost::math::cdf;
try
{
// Print out general info:
cout <<
"_____________________________________________________________\n"
cout << // Print out general info:
"_____________________________________________________________\n"
"Estimated sample sizes required for various confidence levels\n"
"_____________________________________________________________\n\n";
cout << setprecision(5);
@@ -211,16 +269,10 @@ void chi_squared_sample_sized(
std::cout <<
"\n""Message from thrown exception was:\n " << e.what() << std::endl;
}
}
//Message from thrown exception was: for alpha 0.5
// Error in function boost::math::tools::bracket_and_solve_root<double>:
//Unable to bracket root, last nearest value was 1.2446030555722283e-058
} // chi_squared_sample_sized
int main()
{
//
// Run tests for Gear data
// see http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm
// Tests measurements of gear diameter.
@@ -242,21 +294,24 @@ int main()
chi_squared_sample_sized(13.97 * 13.97 - 100, 100);
chi_squared_sample_sized(55, 100);
chi_squared_sample_sized(1, 100);
// List confidence interval multipliers for standard deviation
// for a range of numbers of observations from 2 to a million,
// and for a few alpha values, 0.1, 0.05, 0.01 for condfidences 90, 95, 99 %
confidence_limits_on_std_deviation_alpha(1., 0.1);
confidence_limits_on_std_deviation_alpha(1., 0.05);
confidence_limits_on_std_deviation_alpha(1., 0.01);
return 0;
}
/*
Output:
________________________________________________
2-Sided Confidence Limits For Standard Deviation
________________________________________________
Number of Observations = 100
Standard Deviation = 0.006278908
_____________________________________________
Confidence Lower Upper
Value (%) Limit Limit
@@ -269,45 +324,36 @@ _____________________________________________
99.900 0.00507 0.00812
99.990 0.00489 0.00855
99.999 0.00474 0.00895
______________________________________________
Chi Squared test for sample standard deviation
______________________________________________
Number of Observations = 100
Sample Standard Deviation = 0.00628
Expected True Standard Deviation = 0.10000
Test Statistic = 0.39030
CDF of test statistic: = 1.438e-099
Upper Critical Value at alpha: = 1.232e+002
Upper Critical Value at alpha/2: = 1.284e+002
Lower Critical Value at alpha: = 7.705e+001
Lower Critical Value at alpha/2: = 7.336e+001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard Deviation != 0.100 NOT REJECTED
Standard Deviation < 0.100 NOT REJECTED
Standard Deviation > 0.100 REJECTED
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Variance = 0.10000
Difference to detect = 0.09372
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(lower one (upper one
(lower one- (upper one-
sided test) sided test)
_______________________________________________________________
50.000 2 2
66.667 2 5
75.000 2 10
90.000 4 32
95.000 5 52
@@ -315,15 +361,11 @@ _______________________________________________________________
99.900 13 178
99.990 18 257
99.999 23 337
________________________________________________
2-Sided Confidence Limits For Standard Deviation
________________________________________________
Number of Observations = 10
Standard Deviation = 13.9700000
_____________________________________________
Confidence Lower Upper
Value (%) Limit Limit
@@ -336,45 +378,36 @@ _____________________________________________
99.900 7.69466 42.51593
99.990 7.04085 55.93352
99.999 6.54517 73.00132
______________________________________________
Chi Squared test for sample standard deviation
______________________________________________
Number of Observations = 10
Sample Standard Deviation = 13.97000
Expected True Standard Deviation = 10.00000
Test Statistic = 17.56448
CDF of test statistic: = 9.594e-001
Upper Critical Value at alpha: = 1.692e+001
Upper Critical Value at alpha/2: = 1.902e+001
Lower Critical Value at alpha: = 3.325e+000
Lower Critical Value at alpha/2: = 2.700e+000
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard Deviation != 10.000 REJECTED
Standard Deviation < 10.000 REJECTED
Standard Deviation > 10.000 NOT REJECTED
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Variance = 100.00000
Difference to detect = 95.16090
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(lower one (upper one
(lower one- (upper one-
sided test) sided test)
_______________________________________________________________
50.000 2 2
66.667 2 5
75.000 2 10
90.000 4 32
95.000 5 51
@@ -382,22 +415,19 @@ _______________________________________________________________
99.900 11 174
99.990 15 251
99.999 20 330
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Variance = 100.00000
Difference to detect = 55.00000
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(lower one (upper one
(lower one- (upper one-
sided test) sided test)
_______________________________________________________________
50.000 2 2
66.667 4 10
75.000 8 21
90.000 23 71
95.000 36 115
@@ -405,22 +435,19 @@ _______________________________________________________________
99.900 123 401
99.990 177 580
99.999 232 762
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Variance = 100.00000
Difference to detect = 1.00000
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(lower one (upper one
(lower one- (upper one-
sided test) sided test)
_______________________________________________________________
50.000 2 2
66.667 14696 14993
75.000 36033 36761
90.000 130079 132707
95.000 214283 218612
@@ -428,6 +455,98 @@ _______________________________________________________________
99.900 756333 771612
99.990 1095435 1117564
99.999 1440608 1469711
________________________________________________
2-Sided Confidence Limits For Standard Deviation
________________________________________________
Confidence level (two-sided) = 0.1000000
Standard Deviation = 1.0000000
_____________________________________________
Observations Lower Upper
Limit Limit
_____________________________________________
2 0.5102 15.9472
3 0.5778 4.4154
4 0.6196 2.9200
5 0.6493 2.3724
6 0.6720 2.0893
7 0.6903 1.9154
8 0.7054 1.7972
9 0.7183 1.7110
10 0.7293 1.6452
15 0.7688 1.4597
20 0.7939 1.3704
30 0.8255 1.2797
40 0.8454 1.2320
50 0.8594 1.2017
60 0.8701 1.1805
100 0.8963 1.1336
120 0.9045 1.1203
1000 0.9646 1.0383
10000 0.9885 1.0118
50000 0.9948 1.0052
100000 0.9963 1.0037
1000000 0.9988 1.0012
________________________________________________
2-Sided Confidence Limits For Standard Deviation
________________________________________________
Confidence level (two-sided) = 0.0500000
Standard Deviation = 1.0000000
_____________________________________________
Observations Lower Upper
Limit Limit
_____________________________________________
2 0.4461 31.9102
3 0.5207 6.2847
4 0.5665 3.7285
5 0.5991 2.8736
6 0.6242 2.4526
7 0.6444 2.2021
8 0.6612 2.0353
9 0.6755 1.9158
10 0.6878 1.8256
15 0.7321 1.5771
20 0.7605 1.4606
30 0.7964 1.3443
40 0.8192 1.2840
50 0.8353 1.2461
60 0.8476 1.2197
100 0.8780 1.1617
120 0.8875 1.1454
1000 0.9580 1.0459
10000 0.9863 1.0141
50000 0.9938 1.0062
100000 0.9956 1.0044
1000000 0.9986 1.0014
________________________________________________
2-Sided Confidence Limits For Standard Deviation
________________________________________________
Confidence level (two-sided) = 0.0100000
Standard Deviation = 1.0000000
_____________________________________________
Observations Lower Upper
Limit Limit
_____________________________________________
2 0.3562 159.5759
3 0.4344 14.1244
4 0.4834 6.4675
5 0.5188 4.3960
6 0.5464 3.4848
7 0.5688 2.9798
8 0.5875 2.6601
9 0.6036 2.4394
10 0.6177 2.2776
15 0.6686 1.8536
20 0.7018 1.6662
30 0.7444 1.4867
40 0.7718 1.3966
50 0.7914 1.3410
60 0.8065 1.3026
100 0.8440 1.2200
120 0.8558 1.1973
1000 0.9453 1.0609
10000 0.9821 1.0185
50000 0.9919 1.0082
100000 0.9943 1.0058
1000000 0.9982 1.0018
*/