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I need to fit a model using MLE with Frank copula by linking two discrete univate distribution function $u = F(x)$ and $v = F(y)$ together, and the joint distribution function is

$$ \Phi(x,y) = C(F(x),G(y)) $$

where

$$ C(u,v) = \frac{1}{\theta} \log\!\left( 1+\frac{(\exp(-\theta u)-1)(\exp(-\theta v)-1)}{\exp(-\theta)-1} \right) $$

and end up with the copula density as

$$\begin{align} c(F(x),G(y)) = &~C(F(x),G(y)) - C(F(x-1),G(y)) - C(F(x),G(y-1)) \\ & + C(F(x-1),G(y-1)) \end{align}$$

Therefore, I have to evaluate $\log c(u,v)$ and ultimately I have to evaluate $C$ and its first derivatives $C_u$, $C_v$ and $C_{\theta}$.

I couldn't get it work with my optimization algorithm (I am using fmin_bfgs and fmin_l_bfgs_b from scipy).

Most of the time, fmin_bfgs try to evaluate some very big point and make my code overflow (get -inf) and end with incorrect result.

I tried to use fmin_l_bfgs_b so that I can put bounds on the variable to prevent overflow, but I end up to only make it works (not overflow) in a too small interval, $[-2,2]$ or $[-1,1]$ in some cases. The resulting point will be on the bound and it must not be the correct solution.

I guess the problem is that $\log c$, $C_u$, $C_v$ and $C_{\theta}$ involve too much exp and log and I lost too much precision similar to the case like $\exp(x)-1$ and $\log(1+x)$.

Are there any better method to compute $C$, $C_u$, $C_v$ and $C_{\theta}$ accurately? (or compute $\log c$ and its derivatives directly?)

I am computing them using:

def frank_copula(u,v,t):
    if t == 0.:
        return u*v
    else:
        x1 = expm1(-u*t)
        y1 = expm1(-v*t)
        z1 = expm1(-t)
        return log1p(x1*y1/z1)/(-t)

def frank_copula_u(u,v,t):
    if t == 0.:
        return v
    else:
        x = exp(t*u)
        y = exp(t*v)
        y1 = expm1(t*v)
        z = exp(t)
        return z*y1/(z*(x+y1) - x*y)

def frank_copula_v(u,v,t):
    if t == 0.:
        return u
    else:
        x = exp(t*u)
        x1 = expm1(t*u)
        y = exp(t*v)
        z = exp(t)
        return z*x1/(z*(x1+y) - x*y)

def frank_copula_t(u,v,t):
    if t == 0.: # central difference here
        h = 1e-8
        return (frank_copula(u,v,t+h) - frank_copula(u,v,t-h))/(2.*h)
    else:
        x = exp(-t*u)
        y = exp(-t*v)
        z = exp(-t)
        x1 = expm1(-t*u)
        y1 = expm1(-t*v)
        z1 = expm1(-t)
        tmp1 = -z1*(x1*y*v + y1*x*u) + x1*y1*z
        tmp2 = z1*z1 + x1*y1*z1
        tmp3 = log1p(x1*y1/z1)/(-t)
        return (tmp1/tmp2 + tmp3)/(-t)

EDIT

I fixed the dependency parameter $\theta$ first, and then find out the parameter of $F(x)$ and $G(y)$ together by maximizing $\log L(p) = \sum \log c_{\theta}(F(x|p),G(y|p))$. They involve the parameter $p$ of dimension 2 to 5.

Here is the output of how fmin_l_bfgs_b algorithm evaluate my (minus) log-likelihood function using my workaround, set the bounds to $[-0.2,1.2]$ and $[-1,1]$:

L: array([ 0.,  0.,  0.,  0.,  0.]) => 769.446206148
dL: array([ 0.,  0.,  0.,  0.,  0.]) => array([-338.93276474, -114.42079214, -216.55003298, -137.12475499,
        -52.57668841])
L: array([ 1.2,  1.2,  1.2,  1. ,  1. ]) => 8019.20848421
dL: array([ 1.2,  1.2,  1.2,  1. ,  1. ]) => array([ 10020.77504911,   6095.3498353 ,   8824.13266013,   6763.34035373,
         2021.60815303])
L: array([ 0.31272257,  0.31272257,  0.31272257,  0.26060214,  0.26060214]) => 638.623452278
dL: array([ 0.31272257,  0.31272257,  0.31272257,  0.26060214,  0.26060214]) => array([-77.88688381,  54.50079579,  11.01938969,  98.73509615,  78.99317652])
L: array([ 0.41609651,  0.21822845,  0.28865935,  0.12751238,  0.13439021]) => 612.115679161
dL: array([ 0.41609651,  0.21822845,  0.28865935,  0.12751238,  0.13439021]) => array([-92.53682091,  12.87793807, -22.45693087,  26.74828638,  25.65511475])
L: array([  7.05680743e-01,   1.28761254e-01,   3.21734086e-01,
        -3.25757720e-02,   4.93345936e-04]) => 590.768526126
dL: array([  7.05680743e-01,   1.28761254e-01,   3.21734086e-01,
        -3.25757720e-02,   4.93345936e-04]) => array([-15.61044704,  16.3196277 ,  16.76613757,  10.17553022,   1.73461386])
L: array([ 0.81933183,  0.05230577,  0.26232909, -0.07694669, -0.01632839]) => 587.63001378
dL: array([ 0.81933183,  0.05230577,  0.26232909, -0.07694669, -0.01632839]) => array([-10.36049322,   5.82433346,  10.12673749,  -7.60549293,  -3.70091331])
L: array([ 0.93612938, -0.03119036,  0.15964292, -0.05604783, -0.02092306]) => 585.655142497
dL: array([ 0.93612938, -0.03119036,  0.15964292, -0.05604783, -0.02092306]) => array([-1.60415082,  1.31457884,  5.95207277, -9.31825269, -5.19784526])
L: array([ 0.99490642, -0.09547364,  0.05431211,  0.01089551,  0.01278564]) => 584.829795091
dL: array([ 0.99490642, -0.09547364,  0.05431211,  0.01089551,  0.01278564]) => array([-0.44946283, -0.58748774, -1.01623299, -2.83092327,  1.18001099])
L: array([ 0.99380363, -0.10055615,  0.0527463 ,  0.03080186, -0.00557434]) => 584.840769319
dL: array([ 0.99380363, -0.10055615,  0.0527463 ,  0.03080186, -0.00557434]) => array([ 0.29741289,  0.67052685, -0.11044264,  1.88188906, -3.31504582])
L: array([ 0.99442739, -0.09768137,  0.05363196,  0.01954239,  0.00481047]) => 584.813957801
dL: array([ 0.99442739, -0.09768137,  0.05363196,  0.01954239,  0.00481047]) => array([-0.13879393, -0.04846326, -0.63399355, -0.79472897, -0.7858597 ])
L: array([ 0.99245896, -0.09809096,  0.05491007,  0.02184427,  0.00701989]) => 584.811245584
dL: array([ 0.99245896, -0.09809096,  0.05491007,  0.02184427,  0.00701989]) => array([ 0.16175504,  0.3075338 , -0.23277139, -0.09032161, -0.17666573])
L: array([ 0.99043781, -0.09901399,  0.056742  ,  0.02308671,  0.00811308]) => 584.810330203
dL: array([ 0.99043781, -0.09901399,  0.056742  ,  0.02308671,  0.00811308]) => array([ 0.08694667,  0.38532343, -0.11389679,  0.14966267,  0.06054983])
L: array([ 0.98934696, -0.10209532,  0.05898625,  0.02414812,  0.00863789]) => 584.809284488
dL: array([ 0.98934696, -0.10209532,  0.05898625,  0.02414812,  0.00863789]) => array([ 0.04661414,  0.30695628, -0.01908606,  0.20837613,  0.14123964])
L: array([ 0.9893386 , -0.10784629,  0.06164861,  0.02504006,  0.00849419]) => 584.808428091
dL: array([ 0.9893386 , -0.10784629,  0.06164861,  0.02504006,  0.00849419]) => array([ 0.00282033,  0.01810351,  0.00812252,  0.0266529 ,  0.02348458])
L: array([ 0.98941539, -0.10801042,  0.06168044,  0.02499464,  0.00841819]) => 584.80842517
dL: array([ 0.98941539, -0.10801042,  0.06168044,  0.02499464,  0.00841819]) => array([-0.00429622, -0.00085152, -0.00194202,  0.00036292,  0.0010925 ])

and here is the output if I use my original bounds $[-5,5]$ (this bound works if I treat $F(x)$ and $G(y)$ as independent):

(Remarks: If I treat them as independent, I just need to evaluate the density function $f$,$g$ and they are much more stable and therefore I guess my problem is due to inaccuracy)

L: array([ 0.,  0.,  0.,  0.,  0.]) => 769.446206148
dL: array([ 0.,  0.,  0.,  0.,  0.]) => array([-338.93276474, -114.42079214, -216.55003298, -137.12475499,
        -52.57668841])
L: array([ 5.,  5.,  5.,  5.,  5.]) => inf
dL: array([ 5.,  5.,  5.,  5.,  5.]) => array([ nan,  nan,  nan,  nan,  nan])
L: array([ 0.,  0.,  0.,  0.,  0.]) => 769.446206148
dL: array([ 0.,  0.,  0.,  0.,  0.]) => array([-338.93276474, -114.42079214, -216.55003298, -137.12475499,
        -52.57668841])

and sorry I just found that fmin_bfgs works and will converge to the same point got by using my workaround method. But it will evaluate some big point and get -inf and nan:

dL: array([ 0.,  0.,  0.,  0.,  0.]) => array([-338.93276474, -114.42079214, -216.55003298, -137.12475499,
        -52.57668841])
L: array([ 0.,  0.,  0.,  0.,  0.]) => 769.446206148
L: array([ 338.93276474,  114.42079214,  216.55003298,  137.12475499,
         52.57668841]) => inf
dL: array([ 338.93276474,  114.42079214,  216.55003298,  137.12475499,
         52.57668841]) => array([ nan,  nan,  nan,  nan,  nan])
L: array([  3.38932765e-06,   1.14420792e-06,   2.16550033e-06,
         1.37124755e-06,   5.25766884e-07]) => 769.444241862
dL: array([  3.38932765e-06,   1.14420792e-06,   2.16550033e-06,
         1.37124755e-06,   5.25766884e-07]) => array([-338.93116156, -114.42016384, -216.54896297, -137.12398571,
        -52.57656892])
L: array([ 0.00238973,  0.00080675,  0.00152684,  0.00096683,  0.00037071]) => 768.063616272
dL: array([ 0.00238973,  0.00080675,  0.00152684,  0.00096683,  0.00037071]) => array([-337.80294205, -113.97709305, -215.79496551, -136.58082878,
        -52.49109471])
L: array([ 169.46757724,   57.21079945,  108.27577991,   68.56286091,
         26.28852956]) => inf
dL: array([ 169.46757724,   57.21079945,  108.27577991,   68.56286091,
         26.28852956]) => array([ nan,  nan,  nan,  nan,  nan])
L: array([ 0.00238973,  0.00080675,  0.00152684,  0.00096683,  0.00037071]) => 768.063616272
dL: array([ 0.00238973,  0.00080675,  0.00152684,  0.00096683,  0.00037071]) => array([-337.80294205, -113.97709305, -215.79496551, -136.58082878,
        -52.49109471])
L: array([ 338.93276474,  114.42079214,  216.55003298,  137.12475499,
         52.57668841]) => inf
L: array([ 169.46638237,   57.21039607,  108.27501649,   68.56237749,
         26.28834421]) => inf
L: array([ 84.73319119,  28.60519803,  54.13750824,  34.28118875,  13.1441721 ]) => inf
L: array([ 42.36659559,  14.30259902,  27.06875412,  17.14059437,   6.57208605]) => inf
L: array([ 21.1832978 ,   7.15129951,  13.53437706,   8.57029719,   3.28604303]) => inf
L: array([ 10.5916489 ,   3.57564975,   6.76718853,   4.28514859,   1.64302151]) => inf
L: array([ 5.29582445,  1.78782488,  3.38359427,  2.1425743 ,  0.82151076]) => inf
L: array([ 2.64791222,  0.89391244,  1.69179713,  1.07128715,  0.41075538]) => inf
L: array([ 1.32395611,  0.44695622,  0.84589857,  0.53564357,  0.20537769]) => 2010.68946691
L: array([ 0.25288777,  0.08537274,  0.16157439,  0.10231284,  0.03922902]) => 651.514679659
dL: array([ 0.25288777,  0.08537274,  0.16157439,  0.10231284,  0.03922902]) => array([-214.23525183,  -56.37608777, -123.86215026,  -59.8381013 ,
        -29.35530006])
L: array([ 2.45189744, -1.10216675, -0.00860584, -2.25562141, -0.0889141 ]) => 1140.89767258
dL: array([ 2.45189744, -1.10216675, -0.00860584, -2.25562141, -0.0889141 ]) => array([ 200.97116275, -107.57429828,   77.9175913 , -284.63382774,
        222.78613298])
L: array([ 0.60625088, -0.10545527,  0.1342278 , -0.27658817,  0.01863746]) => 641.856514926
dL: array([ 0.60625088, -0.10545527,  0.1342278 , -0.27658817,  0.01863746]) => array([-179.74804571,  -91.13312507, -130.64600443, -147.53451709,
        -47.29257281])
L: array([ 0.94280274, -0.44136287, -0.34555214, -0.04312928,  0.3963997 ]) => 650.978041317
dL: array([ 0.94280274, -0.44136287, -0.34555214, -0.04312928,  0.3963997 ]) => array([-147.33106574,  -91.10716372, -144.5277363 , -121.87325508,
         59.49497911])
L: array([ 0.7272913 , -0.22626399, -0.03832441, -0.19262499,  0.15449914]) => 638.059222532
dL: array([ 0.7272913 , -0.22626399, -0.03832441, -0.19262499,  0.15449914]) => array([-170.21539216,  -91.09157378, -134.59864658, -140.28045749,
        -16.11667578])
...
L: array([ 0.98943719, -0.10800294,  0.06167076,  0.02498195,  0.00840848]) => 584.808425121
dL: array([ 0.98943719, -0.10800294,  0.06167076,  0.02498195,  0.00840848]) => array([ -6.05248630e-08,   2.30028962e-08,   3.00105580e-08,
        -1.12310441e-07,   1.26875060e-08])

There may be a possibility that my gradient function is computed wrongly, therefore, I check the my gradient function againist finite difference:

def approx_gradient(f):
    def wrapped_f(x0):
        n = len(x0)
        G = np.empty((n,),dtype=np.float64)
        h = np.sqrt(np.finfo(np.float64).eps)

        g1 = f(x0)
        for i in xrange(n):
            x1 = np.array(x0,dtype=np.float64)
            x1[i] += h
            g2 = f(x1)
            G[i] = (g2 - g1) / h
        return G
    return wrapped_f

Print the error as:

# r1 is gradient computed by code, r2 is gradient computed by above finite-difference
d = np.log10(np.mean(np.abs(r1-r2)))
print("dL: %s => %s, error: %s\n" % (repr(x0),repr(r1),str(d)))

and the output:

dL: array([ 0.,  0.,  0.,  0.,  0.]) => array([-338.93276474, -114.42079214, -216.55003298, -137.12475499,
        -52.57668841]), error: -1.12866497845
dL: array([ 338.93276474,  114.42079214,  216.55003298,  137.12475499,
         52.57668841]) => array([ nan,  nan,  nan,  nan,  nan]), error: nan
dL: array([  3.38932765e-06,   1.14420792e-06,   2.16550033e-06,
         1.37124755e-06,   5.25766884e-07]) => array([-338.93116156, -114.42016384, -216.54896297, -137.12398571,
        -52.57656892]), error: -1.3444523876
dL: array([ 0.00238973,  0.00080675,  0.00152684,  0.00096683,  0.00037071]) => array([-337.80294205, -113.97709305, -215.79496551, -136.58082878,
        -52.49109471]), error: -1.76425541273
dL: array([ 169.46757724,   57.21079945,  108.27577991,   68.56286091,
         26.28852956]) => array([ nan,  nan,  nan,  nan,  nan]), error: nan
dL: array([ 0.00238973,  0.00080675,  0.00152684,  0.00096683,  0.00037071]) => array([-337.80294205, -113.97709305, -215.79496551, -136.58082878,
        -52.49109471]), error: -1.76425541273
dL: array([ 0.25288777,  0.08537274,  0.16157439,  0.10231284,  0.03922902]) => array([-214.23525183,  -56.37608777, -123.86215026,  -59.8381013 ,
        -29.35530006]), error: -4.13744558124
dL: array([ 2.45189744, -1.10216675, -0.00860584, -2.25562141, -0.0889141 ]) => array([ 200.97116275, -107.57429828,   77.9175913 , -284.63382774,
        222.78613298]), error: 4.55716763394
dL: array([ 0.60625088, -0.10545527,  0.1342278 , -0.27658817,  0.01863746]) => array([-179.74804571,  -91.13312507, -130.64600443, -147.53451709,
        -47.29257281]), error: -3.6510702955
dL: array([ 0.94280274, -0.44136287, -0.34555214, -0.04312928,  0.3963997 ]) => array([-147.33106574,  -91.10716372, -144.5277363 , -121.87325508,
         59.49497911]), error: -2.92020205735
dL: array([ 0.7272913 , -0.22626399, -0.03832441, -0.19262499,  0.15449914]) => array([-170.21539216,  -91.09157378, -134.59864658, -140.28045749,
        -16.11667578]), error: -3.52122115041
dL: array([ 0.87382717, -0.37020575, -0.21805711, -0.03796395,  0.05122782]) => array([-164.76068544,  -94.42547784, -141.97757919, -126.19087232,
        -39.8567946 ]), error: -3.19928906949
dL: array([ 0.87817125, -0.41141175, -0.15890429, -0.01557913,  0.05239629]) => array([-153.02101444,  -89.44357123, -130.9150735 , -117.22771105,
        -36.90142624]), error: -3.0131329785
dL: array([ 0.89554758, -0.57623572,  0.07770701,  0.07396015,  0.05707019]) => array([-98.94618778, -66.39110534, -79.97680149, -75.88852736, -21.65497929]), error: -3.5340970988
dL: array([ 1.31009037,  1.3391422 , -0.80615157, -0.3413817 , -0.09094827]) => array([  88.75622027,  103.66551146,   28.52931607,   59.92216422,
         34.82483184]), error: -3.86721832568
dL: array([ 1.03325948,  0.06005647, -0.2159125 , -0.0640172 ,  0.00789817]) => array([-57.66593306, -27.77908949, -59.01722568, -47.64785975, -13.70147329]), error: -3.21832537521
dL: array([  9.95932857e-01,  -1.40930257e-01,   1.39440628e-01,
         4.05132617e-02,   5.47288338e-04]) => array([ 27.98975787,  12.98469404,  25.68770109,  19.28943637,   5.48862252]), error: -4.19602531575
dL: array([ 0.98974304, -0.10584977,  0.05390259,  0.02224462,  0.00674051]) => array([-3.07471251, -1.62491769, -2.83340373, -2.41904387, -1.23502546]), error: -3.84209274725
dL: array([ 0.98982872, -0.10776008,  0.0609875 ,  0.0246589 ,  0.00848128]) => array([-0.07129277, -0.05980753, -0.10964847, -0.12157649,  0.00033618]), error: -3.57935797704
dL: array([ 0.98941444, -0.10804747,  0.06173506,  0.02499305,  0.00836133]) => array([-0.00140251, -0.00150752,  0.00292727,  0.00098813, -0.01230324]), error: -4.19996237842
dL: array([ 0.98944092, -0.10800269,  0.06166992,  0.0249806 ,  0.00840967]) => array([ 0.00141414,  0.000485  ,  0.00088848,  0.00042683,  0.00068224]), error: -4.05494671838
dL: array([ 0.98943748, -0.10800296,  0.06167061,  0.02498174,  0.00840839]) => array([  2.32462795e-05,  -1.47758964e-05,  -5.04953544e-06,
        -4.29179409e-05,  -1.69174238e-05]), error: -4.30601047846
dL: array([ 0.9894372 , -0.10800295,  0.06167076,  0.02498195,  0.00840848]) => array([  3.89230032e-07,  -8.59830965e-07,  -1.92379457e-07,
        -1.03506122e-06,  -1.88641715e-06]), error: -4.04379091415
dL: array([ 0.98943719, -0.10800294,  0.06167076,  0.02498195,  0.00840848]) => array([ -6.05248630e-08,   2.30028962e-08,   3.00105580e-08,
        -1.12310441e-07,   1.26875060e-08]), error: -3.67971485116

OK. I think my updated question would be:

  1. I still think my gradient does not compute accurately in the box $[-5,5]^n$. I will approximate fisher information using my gradient function, I couldn't get a semi-positive definite information matrix.
  2. Are there any way to avoid BFGS to evaluate some "big" points?
  3. Are there any way to compute the copula density by computing $f$ and $g$ only for discrete margin?
$\endgroup$
3
  • 1
    $\begingroup$ If the question is about computing the function accurately, can you be specific about the parameter values for which it fails? Saying that BFGS fails is a little too ambiguous and is not easily reproducible. $\endgroup$
    – Kirill
    Oct 8 '14 at 12:42
  • $\begingroup$ Would an equivalent method for fitting the Frank copula that is not based on numerical optimization be of use to you? $\endgroup$
    – Kirill
    Oct 8 '14 at 12:53
  • $\begingroup$ I added some output of my code. and the second question is No. I need to find out the parameter of $F$ and $G$. $\endgroup$
    – wonghang
    Oct 9 '14 at 4:25

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