1{
  2  "config": {
  3    "gamma": 1.0,
  4    "dt": 0.01,
  5    "steps": 60000,
  6    "burn": 30000
  7  },
  8  "bound_sweep": [
  9    {
 10      "lambda": 0.1,
 11      "bound": 1.0,
 12      "max_abs": 0.9999993702580099,
 13      "ratio": 0.9999993702580099
 14    },
 15    {
 16      "lambda": 0.7,
 17      "bound": 1.0,
 18      "max_abs": 0.9567823718178728,
 19      "ratio": 0.9567823718178728
 20    },
 21    {
 22      "lambda": 2.0,
 23      "bound": 1.0,
 24      "max_abs": 0.7876499928340427,
 25      "ratio": 0.7876499928340427
 26    },
 27    {
 28      "lambda": 8.0,
 29      "bound": 1.0,
 30      "max_abs": 0.4914642969231532,
 31      "ratio": 0.4914642969231532
 32    }
 33  ],
 34  "variance_sweep": [
 35    {
 36      "lambda": 0.1,
 37      "emp_var": 0.22184790725174827,
 38      "pred_var": 0.20833333333333334,
 39      "rel_err": 0.06486995480839175
 40    },
 41    {
 42      "lambda": 0.3,
 43      "emp_var": 0.12992748191808579,
 44      "pred_var": 0.15625,
 45      "rel_err": -0.16846411572425102
 46    },
 47    {
 48      "lambda": 0.7,
 49      "emp_var": 0.10295221051962604,
 50      "pred_var": 0.10416666666666667,
 51      "rel_err": -0.01165877901159007
 52    },
 53    {
 54      "lambda": 1.5,
 55      "emp_var": 0.06407518340680937,
 56      "pred_var": 0.0625,
 57      "rel_err": 0.025202934508949948
 58    },
 59    {
 60      "lambda": 3.0,
 61      "emp_var": 0.03290677451371123,
 62      "pred_var": 0.03571428571428571,
 63      "rel_err": -0.07861031361608539
 64    },
 65    {
 66      "lambda": 8.0,
 67      "emp_var": 0.014326501207616789,
 68      "pred_var": 0.014705882352941176,
 69      "rel_err": -0.02579791788205832
 70    }
 71  ],
 72  "covariance_check": {
 73    "lags": [
 74      0.0,
 75      0.1,
 76      0.25,
 77      0.5,
 78      1.0,
 79      2.0,
 80      4.0
 81    ],
 82    "empirical": [
 83      0.37658421035616224,
 84      0.3736366455986064,
 85      0.3605711860031419,
 86      0.3246079799616098,
 87      0.23884382486567812,
 88      0.11436068995098818,
 89      0.016533536598099724
 90    ],
 91    "corrected": [
 92      0.4166666666666667,
 93      0.41397307276201817,
 94      0.401701048522139,
 95      0.36724752063153887,
 96      0.27961901426918007,
 97      0.13402013948545816,
 98      0.02285836534140096
 99    ],
100    "supplied": [
101      -0.0,
102      0.03695748191370174,
103      0.07720072224238693,
104      0.11452641241794159,
105      0.12633591378107903,
106      0.07763043813686953,
107      0.015226849137761721
108    ],
109    "rmse_corrected": 0.03552170395906621,
110    "rmse_supplied": 0.23712879574183154
111  },
112  "kurtosis_sweep": [
113    {
114      "K": 1,
115      "excess_kurtosis": -1.3340502460584176,
116      "variance": 0.4004448444372525,
117      "pred_var": 0.4166666666666667
118    },
119    {
120      "K": 2,
121      "excess_kurtosis": -0.5310430581684686,
122      "variance": 0.3941007196072442,
123      "pred_var": 0.4166666666666666
124    },
125    {
126      "K": 4,
127      "excess_kurtosis": -0.65313176508716,
128      "variance": 0.39936926351644425,
129      "pred_var": 0.4166666666666667
130    },
131    {
132      "K": 8,
133      "excess_kurtosis": 0.058334978098325774,
134      "variance": 0.4070997988505528,
135      "pred_var": 0.4166666666666666
136    },
137    {
138      "K": 16,
139      "excess_kurtosis": -0.22682709084103436,
140      "variance": 0.508631491649338,
141      "pred_var": 0.4166666666666667
142    }
143  ],
144  "quadratic_optimizer": {
145    "none": [
146      0.0,
147      0.0
148    ],
149    "telegraph": [
150      0.0346325191211659,
151      0.028322584889397096
152    ],
153    "ou": [
154      0.03496700678582951,
155      0.025026321910527633
156    ]
157  },
158  "note": "The covariance formula in the prompt is tested literally; it is zero at tau=0 and is not a valid covariance. corrected is derived from the same SDE."
159}