00001 /*************************************************************** 00002 00003 turmon nov 2006 00004 00005 Routines for generating a geometrically-distributed clock, 00006 where we do not care if the clock is larger than M. 00007 That is, if the return value is greater than M, the routine 00008 can choose to not compute it exactly, and return any number 00009 larger than M. 00010 00011 tested new_clock in all three regimes: low q, low p, and moderate 00012 p, corresponding to its three cases. 00013 In the low-q case, I tried q=0.7, 0.5, 0.1, and 0.01 with good 00014 agreement between method 0 and method 2, versus the true density. 00015 In the low p case, I used 00016 q = 1-p = 0.999, M=17, n=1e8, so M*p=0.017 < 1.75e-2, while giving 00017 good contrast between Pr(clk == 1) and Pr(clk == M) as estimated 00018 over n samples (ie, 10.0e4 versus 9.84e4 counts, a difference of 00019 1600 counts in a quantity with standard deviation about 314 counts. 00020 I also did a rough check at q=0.999999, but there is no appreciable 00021 variation of the density across clock value there, so I was not able 00022 to validate the shape of the PDF. 00023 In the moderate-p case, I used q=0.9. This code is identical for 00024 all "mode" inputs, so there's not much to check. 00025 00026 ****************************************************************/ 00027 00028 /* 00029 * this code needs: 00030 * typedef for Clock, currently unsigned 00031 * maximum value CLOCK_MAX (for type Clock), currently UINT_MAX 00032 */ 00033 00034 /* 00035 * get a new clock number based on a geometric(1-q) random variable 00036 * 00037 * this is based on the dead-simple inversion method for the 00038 * exponential random variable, and then quantizing it 00039 * 00040 */ 00041 static 00042 Clock 00043 new_clock_easy(double q, int M) 00044 { 00045 double rng_uniform(); 00046 double clock; /* need the range */ 00047 00048 clock = log(rng_uniform())/log(q) + 1; /* initial choice */ 00049 /* fix it up if needed */ 00050 if (clock > CLOCK_MAX) 00051 clock = CLOCK_MAX; 00052 return((Clock) floor(clock)); 00053 } 00054 00055 /* 00056 * get a new clock number based on a geometric(1-q) random variable 00057 * 00058 * this is the older version of the new_clock routine, which I 00059 * replaced in 2006. It special-cases the low and high q values 00060 * with deterministic values. Because this is a crude way to do 00061 * the approximation, only very extreme values are special-cased. 00062 */ 00063 static 00064 Clock 00065 new_clock_orig(double q, int M) 00066 { 00067 double rng_uniform(); 00068 double clock; /* need the range */ 00069 00070 /* Cheat a little here by handling interval edges, which occur often, 00071 * as special cases, without drawing a new random variable. 00072 * This is somewhat speedier. Size of edges is 10^-8 on each 00073 * side. 00074 * The deterministic approximation only fails on either endpoint 00075 * with probability about 10^-8, so even if all pixels are approximated, 00076 * only about 2 pixels in a 10K x 10K image will be in error. 00077 */ 00078 if (q < 1e-8) 00079 clock = 1.0; 00080 else if (q > (1 - 1e-8)) /* -> log(p) < 1e-8 in magnitude by Taylor */ 00081 clock = CLOCK_MAX; 00082 else { 00083 clock = log(rng_uniform())/log(q) + 1; /* initial choice */ 00084 /* fix it up if needed */ 00085 if (clock > CLOCK_MAX) 00086 clock = CLOCK_MAX; 00087 /* NB, this approximation is compensated for by the "staleness" 00088 * checks within the main loop, that expire all clocks before 00089 * CLOCK_MAX iterations. There is no loss of precision from 00090 * this source. 00091 */ 00092 } 00093 return((Clock) floor(clock)); 00094 } 00095 00096 /* 00097 * Get a new draw X of a random clock 00098 * This is a geometric(p) random variable, where p = 1-q. 00099 * That is, the hold probability is q, and the change probability is p, 00100 * so X is formed by sequences like: 00101 * 00102 * X = 1: C Pr = p 00103 * X = 2: HC Pr = q p 00104 * X = 3: HHC Pr = q^2 p 00105 * X = 4: HHHC Pr = q^3 p 00106 * etc. 00107 * 00108 * In general, Pr(X=k) = p q^k for k=1,... 00109 * And, P(X>k) = q^k, because all such events start out with k "H" letters 00110 * 00111 * We special-case the low and high q values to avoid the rather expensive 00112 * double log() required by the inversion method. The important special 00113 * case is for q near 1, because it is most common. This case requires 00114 * some careful reasoning. The less-common special case is for small 00115 * q, where we just mean q less than 0.5 or so. In this case the 00116 * geometric RV is easily simulated by implementing the Hold/Change 00117 * sequences illustrated above. 00118 */ 00119 static 00120 Clock 00121 new_clock(double q, int M) 00122 { 00123 double rng_uniform(); 00124 const double p = 1-q; 00125 double x; /* the random variable: need the range */ 00126 00127 if (q < 0.75) { 00128 /* CASE 1. q small, or hold time is low. 00129 * Draw by the iteration method. 00130 * The loop below accumulates events of probability q. 00131 */ 00132 /* The constant in the test above affects run time (by 00133 * selecting the fastest generation method) but not accuracy. 00134 * At q = 0.75, on average we draw 1/(1-q) = 4 uniforms, 00135 * about equal to the one uniform plus two logs of the inversion 00136 * method below in CASE 3 (G5, 11/2006). At q=0.2, the code 00137 * below is about 2.2x faster than inversion. At q=0.01, it 00138 * is 3x faster. But note, this is not the dominant case. */ 00139 for (x = 1; rng_uniform() < q && x<=M; x++); 00140 } else if (M*p < 1.75e-2) { 00141 /* CASE 2. q near 1, or p near 0. 00142 * Hold time is high relative to number of iterations 00143 * remaining; typically will return a value X > M. 00144 * This is the most frequent case in our application. 00145 * The code below is a factor of 2.5x faster than the default 00146 * inversion method on a G5. 00147 */ 00148 /* Below, we take one of two branches with probability 00149 * Pr(X>M) = pow(1-p,M) versus Pr(X<=M). 00150 * In the former, overwhelmingly probable case, just return 00151 * CLOCK_MAX. In the latter, generate X conditioned on X<=M. 00152 * 00153 * Because we cannot quickly compute pow(1-p,M), we approximate it 00154 * with the first two terms of its binomial expansion: 00155 * P(X>M) = (1-p)^M ~= 1 - M * p + M(M-1)/2 * p^2 00156 * For M*p < 1.75e-2, the relative accuracy 00157 * A(M,p) = [ 1 - M * p + M(M-1)/2 * p^2 ] / [ (1-p)^M ] 00158 * is better than 1 part in 10^6. For M*p < 3.5e-3, the accuracy 00159 * is better than 1 part in 10^8. 00160 * The test below replaces 00161 * rng_uniform() < 1-rho 00162 * with its equivalent, 00163 * rng_uniform() > rho 00164 * where rho = M * p - M(M-1)/2 * p^2 00165 * = M*p * (1 - 0.5 * (M-1) * p) 00166 * = M*p * (1 - 0.5 * (M * p - p) ) 00167 */ 00168 if (rng_uniform() > M*p*(1 - 0.5*(M*p-p))) { 00169 /* CASE 2a. Generate X | X>M, which is deterministic. */ 00170 /* This branch is taken with probability approximately P(X>M), 00171 * and returns X conditioned on X > M. We almost always 00172 * take this branch since M*p << 1. */ 00173 x = CLOCK_MAX; 00174 } else { 00175 /* CASE 2b. Generate X | X<=M */ 00176 /* This branch is taken with probability approximately P(X<=M), 00177 * and returns X conditioned on X <= M. Censoring the 00178 * realizations with X>M would be one way. But it's easier to 00179 * note that, if X is geometric, then (X-1) % M + 1 is geometric 00180 * conditioned on X<=M. We produce X-1 by taking floor 00181 * and not ceil in the log-of-uniform below. 00182 * We almost never take this branch, so its efficiency is moot. */ 00183 x = fmod(floor(log(rng_uniform())/log(q)), (double) M) + 1; 00184 } 00185 } else { 00186 /* CASE 3. Hold probability is in between: draw by inversion method. */ 00187 x = floor(log(rng_uniform())/log(q)) + 1; /* initial choice */ 00188 /* don't overflow the clock size; note, CLOCK_MAX >= M anyway */ 00189 if (x > CLOCK_MAX) 00190 x = CLOCK_MAX; 00191 } 00192 // mexPrintf("\tp = %g\tclock=%u\n", p, (Clock) x); 00193 // note, x is already an integer > 0 00194 return((Clock) x); 00195 } 00196

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