ARM Architecture

Shibboleths: The Perils of Unvoiced Sibilant Fricatives, Fool Lights, and Different Binary-End result Exams

AS-SALT, JORDAN — Dr. Reza Al-Faisal as soon as had a job supply from Google to work on cutting-edge voice recognition tasks. He turned it down. The 37-year-old Stanford-trained professor of engineering at Al-Balqa’ Utilized College now leads a small cadre of graduate college students in a government-sponsored program to maintain Jordanian society safe from what has now develop into an awesome inflow of refugees from the Palestinian-controlled West Financial institution. “Typically they go to family members right here in Jordan and attempt to keep right here illegally,” he says. “We have already got our personal unemployment issues, particularly among the many youthful technology.”

Al-Faisal’s group has provide you with a technological resolution, using drones and voice recognition. “The voice is the important thing. Native Arabic audio system in West Financial institution and East Financial institution populations have noticeable variations of their use of unvoiced sibilant fricatives.” He factors to a quadcopter drone, its 4 propellers whirring because it hovers in place. “We’ve got put in a high-quality audio system onto these drones, with a speaker and directional microphones. The drone is autonomous. It presents a verbal problem to the suspect. When the suspect solutions, digital sign processing computes the coefficient of palatalization. East Financial institution natives have a imply palatalization coefficient of 0.85. With West Financial institution natives it is just 0.59. If the coefficient is beneath the appropriate threshold, the drone immobilizes the suspect with nonlethal strategies till the closest patrol arrives.”

Considered one of Al-Faisal’s college students presses a button and the drone zooms up and off to the west, embarking on one other pilot run. If Al-Faisal’s system, the SIBFRIC-2000, proves to achieve success in these check runs, it’s seemingly for use on a bigger scale — in order that restricted Jordanian sources can cowl a wider space to patrol for unlawful immigrants. “Two weeks in the past we caught a bunch of eighteen illegals with SIBFRIC. Border Safety would by no means have discovered them. It’s not like in America, the place you’ll be able to discriminate in opposition to individuals as a result of they give the impression of being completely different. East Financial institution, West Financial institution — all of us look the identical. We sound completely different. I’m assured this system will work.”

However some residents of the area are skeptical. “My brother was hit by a drone stunner on certainly one of these exams,” says a 25-year previous bus driver, who didn’t need his title for use for this story. “They mentioned his voice was fallacious. Our household has lived in Jordan for generations. We’re proud residents! How will you belief a machine?”

Al-Faisal declines to present out statistics on what number of circumstances like this have occurred, citing nationwide safety considerations. “The issue is being overstated. Only a few of those incidents have occurred, and in every case the suspect is unhurt. It doesn’t take lengthy for Border Safety to confirm citizenship as soon as they arrive.”

Others say there are rumors of voice coaches in Nablus and Ramallah, serving to determined refugees to beat the system. Al-Faisal is undeterred. “Ha ha ha, that is nonsense. SIBFRIC has an ideal ear; it might hear the slightest nuances of speech. You can not cheat the pc.”

When requested what number of drones are within the pilot program, Al-Faisal demures. “Greater than 5,” he says, “fewer than 5 hundred.” Al-Faisal hopes to ramp up the drone program beginning in 2021.

Disgrace Previous Shtory, Disgrace Previous Shong and Dansh

Does this story sound acquainted? It’d. Right here’s an analogous one from a number of thousand years earlier:

5 And the Gileadites took the passages of Jordan earlier than the Ephraimites: and it was so, that when these Ephraimites which have been escaped mentioned, Let me go over; that the boys of Gilead mentioned unto him, Artwork thou an Ephraimite? If he mentioned, Nay; 6 Then mentioned they unto him, Say now Shibboleth: and he mentioned Sibboleth: for he couldn’t body to pronounce it proper. Then they took him, and slew him on the passages of Jordan: and there fell at the moment of the Ephraimites forty and two thousand.

Judges 12:5-6 (King James Model)

That is the reputed biblical origin of the time period shibboleth, which has come to imply any distinguishing cultural habits that can be utilized to establish a specific group, not only a linguistic one. The Ephraimites couldn’t say SHHHH and it grew to become a useless giveaway of their origin.

On this article, we’re going to speak in regards to the shibboleth and several other different diagnostic exams which have certainly one of two outcomes — cross or fail, sure or no, optimistic or damaging — and a few of the implications of utilizing such a check. And sure, this does influence embedded programs. We’re going to spend fairly a little bit of time taking a look at a selected instance in mathematical phrases. For those who don’t like math, simply skip it each time it will get too “mathy”.

A quantitative shibboleth detector

So let’s say we did need to develop a technological check for separating out individuals who couldn’t say SHHH, and we had some miracle algorithm to guage a “palatalization coefficient” ( P_{SH} ) for unvoiced sibilant fricatives. How would we choose the edge?

Properly, the very first thing to do is attempt to mannequin the system by some means. We took an analogous method in an earlier article on design margin, the place our buddy the Oracle computed likelihood distributions for boiler pressures. Let’s do the identical right here. Suppose the Palestinians (which we’ll name Group 1) have a ( P_{SH} ) which roughly follows a Gaussian distribution with imply μ = 0.59 and commonplace deviation σ = 0.06, and the Jordanians (which we’ll name Group 2) have a ( P_{SH} ) which can be a Gaussian distribution with a imply μ = 0.85 and a normal deviation σ = 0.03.

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
import scipy.stats
from collections import namedtuple

class Gaussian1D(namedtuple('Gaussian1D','mu sigma title id coloration')):
    """ Regular (Gaussian) distribution of 1 variable """
    def pdf(self, x):
        """ probabiliy density operate """
        return scipy.stats.norm.pdf(x,, self.sigma)
    def cdf(self, x):
        """ cumulative distribution operate """
        return scipy.stats.norm.cdf(x,, self.sigma)

d1 = Gaussian1D(0.59, 0.06, 'Group 1 (Palestinian)','P','purple')
d2 = Gaussian1D(0.85, 0.03, 'Group 2 (Jordanian)','J','blue')

def show_binary_pdf(d1, d2, x, fig=None, xlabel=None):
    if fig is None:
        fig = plt.determine(figsize=(6,3))

    for d in (d1,d2):

    if xlabel is just not None:
        ax.set_xlabel(xlabel, fontsize=15)
    ax.set_ylabel('likelihood density')
    return ax
show_binary_pdf(d1,d2,x=np.arange(0,1,0.001), xlabel="$P_{SH}$");

Hmm, now we have a dilemma right here. Each teams have a small risk that the ( P_{SH} ) measurement is within the 0.75-0.8 vary, and this makes it tougher to tell apart. Suppose we determined to set the edge at 0.75. What can be the likelihood of our conclusion being fallacious?

for d in (d1,d2):
    print '%-25s %f' % (d.title, d.cdf(0.75))
Group 1 (Palestinian)     0.996170
Group 2 (Jordanian)       0.000429

Right here the cdf methodology calls the scipy.stats.norm.cdf operate to compute the cumulative distribution operate, which is the likelihood {that a} given pattern from the distribution might be lower than a given quantity. So there’s a 99.617% probability that Group 1’s ( P_{SH} < 0.75 ), and a 0.0429% probability that Group 2’s ( P_{SH} < 0.75 ). One out of each 261 samples from Group 1 will cross the check (although we have been hoping for them to fail) — this is called a false damaging, as a result of a situation that exists (( P_{SH} < 0.75 )) stays undetected. One out of each 2331 samples from Group 2 will fail the check (although we have been hoping for them to cross) — this is called a false optimistic, as a result of a situation that doesn’t exist (( P_{SH} < 0.75 )) is mistakenly detected.

The chances of false optimistic and false damaging are depending on the edge:

for threshold in [0.72,0.73,0.74,0.75,0.76,0.77,0.78]:
    c = [d.cdf(threshold) for d in (d1,d2)]
    print("threshold=%.2f, Group 1 false damaging=%7.5f%%, Group 2 false optimistic=%7.5f%%" 
          % (threshold, 1-c[0], c[1]))
threshold = np.arange(0.7,0.801,0.005)
false_positive = d2.cdf(threshold)
false_negative = 1-d1.cdf(threshold)
determine = plt.determine()
ax = determine.add_subplot(1,1,1)
ax.semilogy(threshold, false_positive, label="false optimistic", coloration="purple")
ax.semilogy(threshold, false_negative, label="false damaging", coloration="blue")
ax.set_xlabel('Threshold $P_{SH}$')
ax.legend(labelspacing=0,loc="decrease proper")
threshold=0.72, Group 1 false damaging=0.01513%, Group 2 false optimistic=0.00001%
threshold=0.73, Group 1 false damaging=0.00982%, Group 2 false optimistic=0.00003%
threshold=0.74, Group 1 false damaging=0.00621%, Group 2 false optimistic=0.00012%
threshold=0.75, Group 1 false damaging=0.00383%, Group 2 false optimistic=0.00043%
threshold=0.76, Group 1 false damaging=0.00230%, Group 2 false optimistic=0.00135%
threshold=0.77, Group 1 false damaging=0.00135%, Group 2 false optimistic=0.00383%
threshold=0.78, Group 1 false damaging=0.00077%, Group 2 false optimistic=0.00982%

If we would like a decrease likelihood of false positives (fewer Jordanians detained for failing the check) we are able to accomplish that by decreasing the edge, however on the expense of elevating the likelihood of false negatives (extra Palestinians unexpectedly passing the check and never detained), and vice-versa.

Components utilized in selecting a threshold

There’s a entire science round binary-outcome exams, primarily within the medical business, involving sensivity and specificity, and it’s not only a matter of likelihood distributions. There are two different points that make an enormous distinction in figuring out a very good check threshold:

  • base price — the likelihood of the situation really being true, generally referred as prevalence in medical prognosis
  • the implications of false positives and false negatives

Each of those are vital as a result of they have an effect on our interpretation of false optimistic and false damaging chances.

Base price

The chances we calculated above are conditional chances — in our instance, we calculated the likelihood that an individual recognized to be from the Palestinian inhabitants handed the SIBFRIC check, and the likelihood that an individual recognized to be from the Jordanian inhabitants failed the SIBFRIC exams.

It’s additionally vital to contemplate the joint likelihood distribution — suppose that we try to detect a really unusual situation. On this case the false optimistic price might be amplified relative to the false damaging price. Let’s say now we have some situation C that has a base price of 0.001, or one in a thousand, and there’s a check with a false optimistic price of 0.2% and a false damaging price of 5%. This appears like a extremely unhealthy check: we must always steadiness the chances by decreasing the false damaging price and permitting a better false optimistic price. The web incidence of false positives for C might be 0.999 × 0.002 = 0.001998, and the online incidence of false negatives might be 0.001 × 0.05 = 0.00005. If we had a million individuals we check for situation C:

  • 1000 even have situation C

    • 950 individuals are accurately identified has having C
    • 50 individuals will stay undetected (false negatives)

  • 999000 don’t even have situation C

    • 997002 individuals are accurately identified as not having C
    • 1998 individuals are incorrectly identified as having C (false positives)

The web false optimistic price is way larger than the online false damaging price, and if we had a distinct check with a false optimistic price of 0.1% and a false damaging price of 8%, this may really be higher, despite the fact that the conditional chances of false positives and false negatives look much more lopsided. This is called the false optimistic paradox.


Let’s proceed with our hypothetical situation C with a base price of 0.001, and the check that has a false optimistic price of 0.2% and a false damaging price of 5%. And suppose that the implications of false positives are pointless hospitalization and the implications of false negatives are sure loss of life:

  • 997002 identified as not having C → reduction
  • 1998 incorrectly identified as having C → pointless hospitalization, monetary value, annoyance
  • 950 accurately identified as having C → therapy, reduction
  • 50 incorrectly identified as not having C → loss of life

If the check could be modified, we’d need to scale back the false damaging, even when it raises the online false optimistic price larger. Would decreasing 50 deaths per million to 10 deaths per million be price it if it raises the false optimistic price of pointless hospitalization from 1998 per million to, say, 5000 per million? 20000 per million?

Penalties can not often be in contrast instantly; extra typically now we have an apples-to-oranges comparability like loss of life vs. pointless hospitalization, or permitting criminals to be free vs. incarcerating the harmless. If we need to deal with a tradeof quantitatively, we’d have to assign some type of metric for the implications, like assigning a worth of $10 million for an pointless loss of life vs. $10,000 for an pointless hospitalization — in such a case we are able to reduce the online anticipated loss over a complete inhabitants. In any other case it turns into an moral query. In jurisprudence there’s the thought of Blackstone’s ratio: “It’s higher that ten responsible individuals escape than that one harmless endure.” However the post-2001 political local weather in the USA appears to be that detaining the harmless is extra fascinating than permitting terrorists or unlawful immigrants to stay at massive. Arithmetic alone gained’t assist us out of those quandaries.

Optimizing a threshold

OK, so suppose now we have a state of affairs that may be described utterly in mathematical phrases:

The anticipated worth over all outcomes is

E[v] &= v_{Aa}P(Aa)+v_{Ab}P(Ab) + v_{Ba}P(Ba) + v_{Bb}P(Bb)cr
&= v_{Aa}p_A P(a | A) cr
&+ v_{Ab}p_A P(b | A) cr
&+ v_{Ba}p_B P(a | B) cr
&+ v_{Bb}p_B P(b | B)

These conditional chances ( P(a | A) ) (denoting the likelihood of the classification ( a ) on condition that the pattern is in ( A )) could be decided with the CDF features; for instance, if the pattern is in A then ( P(a | A) = P(x > x_0 | A) = 1 – P(x le x_0) = 1 – F_A(x_0) ), and as soon as we all know that, then now we have

E[v] &= v_{Aa}p_A (1-F_A(x_0)) cr
&+ v_{Ab}p_A F_A(x_0) cr
&+ v_{Ba}p_B (1-F_B(x_0)) cr
&+ v_{Bb}p_B F_B(x_0) cr
E[v] &= p_A left(v_{Ab} + (v_{Aa}-v_{Ab})left(1-F_A(x_0)proper)proper) cr
&+ p_B left(v_{Bb} + (v_{Ba}-v_{Bb})left(1-F_B(x_0)proper)proper)

( E[v] ) is definitely a operate of the edge ( x_0 ), and we are able to find its most worth by figuring out factors the place its partial by-product ( {partial E[v] over partial {x_0}} = 0: )

$$0 = {partial E[v] over partial {x_0}} = p_A(v_{Ab}-v_{Aa})f_A(x_0) + p_B(v_{Bb}-v_{Ba})f_B(x_0)$$

$${f_A(x_0) over f_B(x_0)} = rho_p rho_v = -frac{p_B(v_{Bb}-v_{Ba})}{p_A(v_{Ab}-v_{Aa})}$$

the place the ( rho ) are ratios for likelihood and for worth tradeoffs:

rho_p &= p_B/p_A cr
rho_v &= -frac{v_{Bb}-v_{Ba}}{v_{Ab}-v_{Aa}}

One fascinating factor about this equation is that since chances ( p_A ) and ( p_B ) and PDFs ( f_A ) and ( f_B ) are optimistic, which means that ( v_{Bb}-v_{Ba} ) and ( v_{Ab}-v_{Aa} ) have to be reverse indicators, in any other case… properly, let’s see:

Case examine: Embolary Pulmonism

Suppose now we have an obscure medical situation; let’s name it an embolary pulmonism, or EP for brief. This have to be handled inside 48 hours, or the affected person can transition in minutes from seeming completely regular, to a situation through which a small portion of their lungs degrade quickly, dissolve, and clog up blood vessels elsewhere within the physique, resulting in excessive discomfort and an nearly sure loss of life. Earlier than this fast decline, the one signs are a sore throat and achy eyes.

We’re growing a cheap diagnostic check ( T_1 ) (let’s suppose it prices $1) the place the affected person seems right into a machine and it takes an image of the affected person’s eyeballs and makes use of machine imaginative and prescient to provide you with some metric ( x ) that may differ from 0 to 100. We have to choose a threshold ( x_0 ) such that if ( x > x_0 ) we diagnose the affected person with EP.

Let’s take into account some math that’s not fairly sensible:

  • situation A: affected person has EP
  • situation B: affected person doesn’t have EP
  • incidence of EP in sufferers complaining of sore throats and achy eyes: ( p_A = ) 0.004% (40 per million)
  • worth of Aa (right prognosis of EP): ( v_{Aa}=- ) $100000
  • worth of Ab (false damaging, affected person has EP, prognosis of no EP): ( v_{Ab}=0 )
  • worth of Ba (false optimistic, affected person doesn’t have EP, prognosis of EP): ( v_{Ba}=- )$5000
  • worth of Bb (right prognosis of no EP): ( v_{Bb} = 0 )

So now we have ( v_{Ab}-v_{Aa} = ) $100000 and ( v_{Bb}-v_{Ba} = ) $5000, for a ( rho_v = -0.05, ) which suggests that we’re on the lookout for a threshold ( x_0 ) the place ( frac{f_A(x_0)}{f_B(x_0)} = -0.05frac{p_B}{p_A} ) is damaging, and that by no means happens with any actual likelihood distributions. Actually, if we glance rigorously on the values ( v ), we’ll see that after we diagnose a affected person with EP, it all the time has a better value: If we accurately diagnose them with EP, it prices $100,000 to deal with. If we incorrectly diagnose them with EP, it prices $5,000, maybe as a result of we are able to run a lung biopsy and another fancy check to find out that it’s not EP. Whereas if we give them a damaging prognosis, it doesn’t value something. This means that we must always all the time choose to present sufferers a damaging prognosis. So we don’t even want to check them!

Affected person: “Hello, Doc, I’ve achy eyes and a sore throat, do I’ve EP?”

Physician: (seems at affected person’s elbows studiously for just a few seconds) “Nope!”

Affected person: (relieved) “Okay, thanks!”

What’s fallacious with this image? Properly, all of the values look cheap besides for 2 issues. First, we haven’t included the $1 value of the eyeball check… however that may have an effect on all 4 outcomes, so let’s simply state that the values ( v ) are along with the price of the check. The extra vital problem is the false damaging, the Ab case, the place the affected person is identified incorrectly as not having EP, and it’s seemingly the affected person will die. Maybe the hospital’s insurance coverage firm has estimated a price of $10 million per case to cowl wrongful loss of life civil fits, through which case we must be utilizing ( v _ {Ab} = -$10^7 ). So right here’s our revised description:

  • situation A: affected person has EP
  • situation B: affected person doesn’t have EP
  • incidence of EP in sufferers complaining of sore throats and achy eyes: ( p_A = ) 0.004% (40 per million)
  • worth of Aa (right prognosis of EP): ( v_{Aa}=- ) $100000 (rationale: imply value of therapy)
  • worth of Ab (false damaging, affected person has EP, prognosis of no EP): ( v_{Ab}=-$10^7 ) (rationale: imply value of ensuing legal responsibility as a consequence of excessive danger of loss of life)
  • worth of Ba (false optimistic, affected person doesn’t have EP, prognosis of EP): ( v_{Ba}=- )$5000 (rationale: imply value of extra exams to substantiate)
  • worth of Bb (right prognosis of no EP): ( v_{Bb} = 0 ) (rationale: no additional therapy wanted)

The equation for selecting ( x_0 ) then turns into

rho_p &= p_B/p_A = 0.99996 / 0.00004 = 24999 cr
rho_v &= -frac{v_{Bb}-v_{Ba}}{v_{Ab}-v_{Aa}} = -5000 / {-9.9}occasions 10^6 approx 0.00050505 cr
{f_A(x_0) over f_B(x_0)} &= rho_prho_v approx 12.6258.

Now we have to know extra about our check. Suppose that the outcomes of the attention machine check have regular likelihood distributions with ( mu=55, sigma=5.3 ) for sufferers with EP and ( mu=40, sigma=5.9 ) for sufferers with out EP.

x = np.arange(0,100,0.1)
dpos = Gaussian1D(55, 5.3, 'A (EP-positive)','A','purple')
dneg = Gaussian1D(40, 5.9, 'B (EP-negative)','B','inexperienced')

show_binary_pdf(dpos, dneg, x, xlabel="check consequence $x$");

Yuck. This doesn’t appear to be an excellent check; there’s quite a lot of overlap between the likelihood distributions.

At any price, suppose we choose a threshold ( x_0=47 ); what sort of false optimistic / false damaging charges will we get, and what’s the anticipated general worth?

from import show, HTML

def analyze_binary(dneg, dpos, threshold):
    Returns confusion matrix
    pneg = dneg.cdf(threshold)
    ppos = dpos.cdf(threshold)
    return np.array([[pneg, 1-pneg],
                     [ppos, 1-ppos]])
def show_binary_matrix(confusion_matrix, threshold, distributions, 
                       outcome_ids, ppos, vmatrix,
    if special_format is None:
        special_format = {}
    def cellinfo(c, p, v):
        # joint likelihood = c*p
        jp = c*p
        return (c, jp, jp*v)
    def rowcalc(i,confusion_row):
        """ write this for rows containing N components, not simply 2 """
        p = ppos if (i == 1) else (1-ppos)
        return [cellinfo(c,p,vmatrix[i][j]) for j,c in enumerate(confusion_row)]
    Jfmtlist = special_format.get('J')
    cfmtlist = special_format.get('c')
    vfmtlist = special_format.get('v')
        if isinstance(vfmtlist, basestring):
            vfmt_general = vfmtlist
            vfmt_general = vfmtlist[0]
    def rowfmt(row,dist):
        def get_format(fmt, icell, default):
            if fmt is None:
                return default
            if isinstance(fmt,basestring):
                return fmt
            return fmt[icell] or default
        def cellfmt(icell):
            Jfmt = get_format(Jfmtlist, icell, '%.7f')
            cfmt = get_format(cfmtlist, icell, '%.5f')
            vfmt = get_format(vfmtlist, icell, '%.3f')
            return '<td>'+cfmt+'<br>J='+Jfmt+'<br>wv='+vfmt+'</td>'
        return '<th>'+dist.title+'</th>' + ''.be a part of(
                  (cellfmt(i) % cell)
                  for i,cell in enumerate(row))

    rows = [rowcalc(i,row) for i,row in enumerate(confusion_matrix)]
    vtot = sum(v for row in rows for c,J,v in row)     
    if not isinstance (threshold, basestring):
        threshold = 'x_0 = %s' % threshold
    return HTML(('<p>Report for threshold (%s rightarrow E[v]=)'
                ''.be a part of('<th>%s</th>' % id for id in outcome_ids)
                +''.be a part of('<tr>%s</tr>' % rowfmt(row,dist) 
                         for row,dist in zip(rows,distributions))

threshold = 47
C = analyze_binary(dneg, dpos, threshold)
show_binary_matrix(C, threshold, [dneg, dpos], 'ba', 40e-6,[[0,-5000],[-1e7, -1e5]],

Report for threshold (x_0 = 47 rightarrow E[v]=)$-618.57

b a
B (EP-negative) 0.88228
A (EP-positive) 0.06559

Right here we’ve proven a modified confusion matrix displaying for every of the 4 outcomes the next portions:

  • Conditional likelihood of every end result: ( start{bmatrix}P(b | B) & P(a | B) cr P(b | A) & P(a | A)finish{bmatrix} ) — learn every entry like ( P(a | B) ) as “the likelihood of ( a ), on condition that ( B ) is true”, in order that the numbers in every row add as much as 1
  • J: Joint likelihood of the result: ( start{bmatrix}P(Bb) & P(Ba) cr P(Ab) & P(Aa)finish{bmatrix} ) — learn every entry like ( P(Ba) ) as “The likelihood that ( B ) and ( a ) are true”, in order that the numbers in all the matrix add as much as 1
  • wv: Weighted contribution to anticipated worth = joint likelihood of the result × its worth

For instance, if the affected person doesn’t have EP, there’s about an 88.2% probability that they are going to be identified accurately and an 11.8% probability that the check will produce a false optimistic. If the affected person does have EP, there’s a couple of 6.6% probability the check will produce a false damaging, and a 93.4% probability the check will accurately diagnose that they’ve EP.

The actually fascinating factor right here is the contribution to anticipated worth. Bear in mind, the false damaging (Ab) is de facto unhealthy, because it has a worth of $10 million, but it surely’s additionally very uncommon due to the low incidence of EP and the truth that condtional likelihood of a false damaging is barely 6.6%. Whereas the main contribution to anticipated worth comes from the false optimistic case (Ba) that happens in nearly 11.8% of the inhabitants.

We must always have the ability to use a better threshold to scale back the anticipated value over the inhabitants:

for threshold in [50, 55]:
    C = analyze_binary(dneg, dpos, threshold)
    show(show_binary_matrix(C, threshold, [dneg, dpos], 'ba', 40e-6,[[0,-5000],[-1e7, -1e5]], 

Report for threshold (x_0 = 50 rightarrow E[v]=)$-297.62

b a
B (EP-negative) 0.95495
A (EP-positive) 0.17274

Report for threshold (x_0 = 55 rightarrow E[v]=)$-229.52

b a
B (EP-negative) 0.99449
A (EP-positive) 0.50000

The optimum threshold ought to in all probability be someplace between ( x_0=50 ) and ( x_0=55 ), since in a single case the contributions to anticipated worth is usually from the false optimistic case, and within the different from the false damaging case. If now we have a very good threshold, these contributions are across the identical order of magnitude from the false optimistic and false damaging circumstances. (They gained’t essentially be equal, although.)

To compute this threshold we’re on the lookout for

$$rho = {f_A(x_0) over f_B(x_0)} = rho_prho_v approx 12.6258.$$

We will both clear up it utilizing numerical strategies, or attempt to clear up analytically utilizing the regular distribution likelihood density

$$f(x) = frac{1}{sigmasqrt{2pi}}e^{-(x-mu)^2/2sigma^2}$$

which provides us

$$frac{1}{sigma_A}e^{-(x_0-mu_A)^2/2sigma_A{}^2} = frac{1}{sigma_B}rho e^{-(x_0-mu_B)^2/2sigma_B{}^2}$$

and taking logs, we get

$$-lnsigma_A-(x_0-mu_A)^2/2sigma_A{}^2 = lnrho -lnsigma_B-(x_0-mu_B)^2/2sigma_B{}^2$$

If we set ( u = x_0 – mu_A ) and ( Delta = mu_B – mu_A ) then we get

$$-u^2/2sigma_A{}^2 = lnrhofrac{sigma_A}{sigma_B} -(u-Delta)^2/2sigma_B{}^2$$

$$-sigma_B{}^2u^2 = 2sigma_A{}^2sigma_B{}^2left(lnrhofrac{sigma_A}{sigma_B} proper) -sigma_A{}^2(u^2 – 2Delta u + Delta^2)$$

which simplifies to ( Au^2 + Bu + C = 0 ) with

A &= sigma_B{}^2 – sigma_A{}^2 cr
B &= 2Deltasigma_A{}^2 cr
C &= 2sigma_A{}^2sigma_B{}^2left(lnrhofrac{sigma_A}{sigma_B}proper) – Delta^2sigma_A{}^2

We will clear up this with the alternate type of the quadratic system ( u = frac{2C}{-B pm sqrt{B^2-4AC}} ) which might compute the basis(s) even with ( A=0 (sigma_A = sigma_B = sigma) ), the place it simplifies to ( u=-C/B=frac{-sigma^2 ln rho}{mu_B – mu_A} + frac{mu_B – mu_A}{2} ) or ( x_0 = frac{mu_B + mu_A}{2} – frac{sigma^2 ln rho}{mu_B – mu_A} ).

def find_threshold(dneg, dpos, ppos, vmatrix):
    num = -(1-ppos)*(vmatrix[0][0]-vmatrix[0][1])
    den = ppos*(vmatrix[1][0]-vmatrix[1][1])
    rho = num/den
    A = dneg.sigma**2 - dpos.sigma**2
    ofs =
    delta = -
    B = 2.0*delta*dpos.sigma**2
    C = (2.0 * dneg.sigma**2 * dpos.sigma**2 * np.log(rho*dpos.sigma/dneg.sigma) 
         - delta**2 * dpos.sigma**2)
    if (A == 0):
        roots = [ofs-C/B]
        D = B*B-4*A*C
        roots = [ofs + 2*C/(-B-np.sqrt(D)),
                 ofs + 2*C/(-B+np.sqrt(D))]
    # Calculate anticipated worth, in order that if now we have a couple of root,
    # the caller can decide which is best
    pneg = 1-ppos
    outcomes = []
    for i,root in enumerate(roots):
        cneg = dneg.cdf(root)
        cpos = dpos.cdf(root)
        Ev =  (cneg*pneg*vmatrix[0][0]
    return outcomes
find_threshold(dneg, dpos, 40e-6, [[0,-5000],[-1e7, -1e5]])
[(182.23914143860179, -400.00000000000006),
 (53.162644275683974, -212.51747111423805)]
threshold =53.1626
for x0 in [threshold, threshold-0.1, threshold+0.1]:
    C = analyze_binary(dneg, dpos, x0)
    show(show_binary_matrix(C, x0, [dneg, dpos], 'ba', 40e-6,[[0,-5000],[-1e7, -1e5]],

Report for threshold (x_0 = 53.1626 rightarrow E[v]=)$-212.52

b a
B (EP-negative) 0.98716
A (EP-positive) 0.36442

Report for threshold (x_0 = 53.0626 rightarrow E[v]=)$-212.58

b a
B (EP-negative) 0.98659
A (EP-positive) 0.35735

Report for threshold (x_0 = 53.2626 rightarrow E[v]=)$-212.58

b a
B (EP-negative) 0.98771
A (EP-positive) 0.37153

So it seems like we’ve discovered the edge ( x_0 = 53.0626 ) to maximise anticipated worth of all doable outcomes at $-212.52. The overwhelming majority (98.72%) of individuals taking the check don’t incur any value or hassle past that of the check itself.

Nonetheless, it’s considerably unsatisfying to have such a excessive false damaging price: over 36% of sufferers who’re EP-positive are undetected by our check, and are prone to die. To place this into perspective, take into account 1,000,000 sufferers who take this check. The anticipated variety of them for every end result are

  • 12842 might be identified with EP however are literally EP-negative (false optimistic) and require $5000 in exams to substantiate
  • 15 might be EP-positive however won’t be identified with EP (false damaging) and prone to die
  • 25 might be EP-positive and accurately identified and incur $100,000 in therapy
  • the remainder are EP-negative and accurately identified.

That doesn’t appear truthful to these 15 individuals, simply to assist scale back the false optimistic price.

We might attempt skewing the check by assigning a worth of $100 million, slightly than $10 million, for the false damaging case, as a result of it’s actually actually unhealthy:

find_threshold(dneg, dpos, 40e-6, [[0,-5000],[-1e8, -1e5]])
[(187.25596232479654, -4000.0000000000005),
 (48.145823389489138, -813.91495238472135)]
threshold = 48.1458
for x0 in [threshold, threshold-0.1, threshold+0.1]:
    C = analyze_binary(dneg, dpos, x0)
    show(show_binary_matrix(C, x0, [dneg, dpos], 'ba', 40e-6,[[0,-5000],[-1e8, -1e5]],

Report for threshold (x_0 = 48.1458 rightarrow E[v]=)$-813.91

b a
B (EP-negative) 0.91631
A (EP-positive) 0.09796

Report for threshold (x_0 = 48.0458 rightarrow E[v]=)$-814.23

b a
B (EP-negative) 0.91367
A (EP-positive) 0.09474

Report for threshold (x_0 = 48.2458 rightarrow E[v]=)$-814.23

b a
B (EP-negative) 0.91888
A (EP-positive) 0.10126

There, by transferring the edge downward by about 5 factors, we’ve decreased the false damaging price to simply below 10%, and the false optimistic price is simply over 8%. The anticipated worth has almost quadrupled, although, to $-813.91, principally as a result of the false optimistic price has elevated, but additionally as a result of the price of a false damaging is way larger.

Now, for each million individuals who take the check, we might anticipate

  • round 83700 may have a false optimistic
  • 36 might be accurately identified with EP
  • 4 will incorrectly stay undiagnosed and sure die.

Someway that doesn’t sound very satisfying both. Can we do any higher?

Fool lights

Let’s put apart our dilemma of selecting a prognosis threshold, for a couple of minutes, and discuss fool lights. This time period typically refers to indicator lights on automotive instrument panels, and apparently confirmed up in print round 1960. A July 1961 article by Phil McCafferty in Fashionable Science, referred to as Let’s Deliver Again the Lacking Gauges, states the difficulty succinctly:

Automobile makers have you ever and me discovered for idiots — to the tune of about eight million {dollars} a yr. That is the estimated quantity that auto makers save by promoting us little blink-out bulbs — recognized to fervent nonbelievers as “fool lights” — on 5 million new automobiles as an alternative of becoming them out with extra significant gauges.

Not the whole lot about blink-outs is unhealthy. They do give a conspicuous warning when one thing is critically amiss. However they don’t inform sufficient, or inform it quickly sufficient to be wholly dependable. That automobile patrons aren’t joyful is attested by the truth that gauge makers gleefully promote some quarter of 1,000,000 accent devices a yr to individuals who insist on understanding what’s occurring below their hoods.

He goes on to say:

There’s little comfort in being advised your engine has overheated after it’s executed it. With these great old-time gauges, a climbing needle gave you warning earlier than you bought into hassle.

The essential blink-out precept is opposite to all guidelines of security.
Blink-outs don’t “fail protected.” The system operates on the peace of mind
that each one is properly when the lights are off. If a bulb burns out whereas
you’re touring, you’ve misplaced your warning system.

Certain, most indicators stay on momentarily throughout beginning, which
makes it doable to examine them for burnouts. However this has been recognized to have issues.

Take into account the Midwestern husband who rigorously lectured his younger spouse
on the significance of watching gauges, then traded within the previous automobile for a
new blink-out mannequin. Anxious to attempt it out, the spouse fired it up
the minute it arrived with out ready for her husband to come back residence.
Panicked by three evident lights, she assumed the worst, threw
open the hood, and was met by the odor of latest engine paint burning.
With out hesitation, she popped off the cap and crammed the crank-case
to the brim—with water.

The massive blink-out gripe is that the lights fail to inform the
diploma of what’s happening. Most oil-pressure blink-outs
flip off at about 10 to fifteen kilos’ strain. But this isn’t almost
sufficient to lubricate an engine at 70 m.p.h.

The generator blink-out, in contrast to an ammeter, tells solely whether or not
or not the generator is producing present, not how a lot.
You could be heading for a useless battery in case you are utilizing extra
present than the generator is producing. A battery can
even be ruined by pouring an extreme cost into it,
and over-production can kill a generator. But in all circumstances
the generator is working, so the sunshine is off.

The lights disguise the secrets and techniques {that a} sinking, climbing,
or fluctuating needle can reveal. Due to this, blink-outs
are one of many biggest issues that ever occurred to a
shady used-car vendor.

McCafferty appears to have been a little bit of a zealot on this matter, publishing a November 1955 article in Fashionable Science, I Just like the Gauges Detroit Left Out, although it doesn’t point out the time period “fool mild”. The earliest use of “fool mild” in frequent media appears to be within the January 1960 problem of Fashionable Science, protecting some automotive equipment like this volt-ammeter equipment:

Apparently sufficient, on the earlier web page is an article (TACHOMETER: You Can Assemble One Your self with This $14.95 Equipment) displaying a schematic and set up of a tachometer circuit that will get its enter from one of many distributor terminals, and makes use of a PNP transistor to ship a cost pulse to a capacitor and ammeter everytime the automobile’s distributor sends a high-voltage pulse to the spark plugs:

The earliest use of “fool mild” I might discover in any publication is from a 1959 monograph on Customary Cells from the Nationwide Bureau of Requirements which states

It’s realized that the usage of good-bad indicators have to be approached with warning. Good-bad lights, typically disparagingly known as fool lights, are incessantly resented by the technician due to the plain implication. Moreover, expert technicians could really feel {that a} good-bad indication doesn’t give them sufficient data to assist their intuitions. Nevertheless, when all elements are weighed, good-bad indicators seem to finest match the necessities for a sign means which may be interpreted shortly and precisely by all kinds of personnel whether or not educated or untrained.

Whatever the time period’s historical past, we’re caught with these lights in lots of circumstances, they usually’re a compromise between two rules:

  • fool lights are an efficient and cheap methodology of catching the operator’s consideration to certainly one of many doable situations that may happen, however they disguise data by lowering a steady worth to a real/false indication
  • a numeric consequence, proven by a dial-and-needle gauge or a numeric show, can present extra helpful data than an fool mild, however it’s dearer, doesn’t draw consideration as simply as an fool mild, and it requires the operator to interpret the numeric worth

¿Por qué no los dos?

If we don’t have an ultra-cost-sensitive system, why not have each? Computerized screens are quite common nowadays, and it’s comparatively simple to show each a PASS/FAIL or YES/NO indicator — for drawing consideration to a doable downside — and a worth that enables the operator to interpret the information.

Since 2008, new automobiles bought in the USA have been required to have a tire strain monitoring system (TPMS). As a driver, I each find it irresistible and hate it. The TPMS is nice for coping with gradual leaks earlier than they develop into an issue. I carry a small electrical tire pump that plugs into the 12V socket in my automobile, so if the TPMS mild comes on, I can pull over, examine the tire strain, and pump up the one which has a leak to its regular vary. I’ve had gradual leaks which have lasted a number of weeks earlier than they begin getting worse. Or generally they’re simply low strain as a result of it’s November or December and the temperature has decreased. What I don’t like is that there’s no numerical gauge. If my TPMS mild comes on, I’ve no solution to distinguish a slight lower in tire strain (25psi vs. the conventional 30psi) vs. a dangerously low strain (15psi), until I cease and measure all 4 tires with a strain gauge. I’ve no solution to inform how shortly the tire strain is lowering, so I can determine whether or not to maintain driving residence and cope with it later, or whether or not to cease on the nearest doable service station. It might be nice if my automobile had an data display the place I might learn the tire strain readings and determine what to do primarily based on having that data.

So far as medical diagnostic exams go, utilizing the additional data from a uncooked check rating could be a tougher resolution, particularly in circumstances the place the possibilities and prices of each false positives and false negatives are excessive. Within the EP instance we checked out earlier, we had a 0-to-100 check with a threshold of someplace within the 55-65 vary. Permitting a health care provider to make use of their judgment when deciphering this type of a check may be a very good factor, particularly when the physician can attempt to make use of different data. However as a affected person, how am I to know? If I’m getting examined for EP and I’ve a 40 studying, my physician could be very assured that I don’t have EP, whereas with a 75 studying, it’s a no brainer to begin therapy immediately. However these numbers close to the edge are difficult.

Triage (¿Por qué no los tres?)

In drugs, the time period triage refers to a strategy of fast prioritization or categorization with the intention to decide which sufferers must be served first. The concept is to attempt to take advantage of distinction, given restricted sources — so sufferers who’re sick or injured, however not in any quick hazard, could have to attend.

As an engineer, my colleagues and I exploit triage as a solution to categorize points in order that we are able to focus solely on the few which can be most vital. A few occasions a yr we’ll go over the unresolved points in our problem monitoring database, to determine which we’ll deal with within the close to time period. One of many issues I’ve seen is that our points fall into three sorts:

  • Points that are clearly low precedence — these are ones that we are able to take a look at in just a few seconds and agree, “Oh, yeah, we don’t like that habits but it surely’s only a minor annoyance and isn’t going to trigger any actual hassle.”
  • Points that are clearly excessive precedence — these are ones that we are able to additionally take a look at in just a few seconds and agree that we have to deal with them quickly.
  • Points with uncertainty — we take a look at these and type of sit and stare for some time, or have arguments inside the group, about whether or not they’re vital or not.

Those within the final class take quite a lot of time, and gradual this course of down immensely. I might a lot slightly come to a 30-second consensus of L/H/U (“low precedence”/”excessive precedence”/”unsure”) and get via the entire listing, then come again and undergo the U points one after the other at a later date.

Let’s return to our EP case, and use the outcomes of our $1 eyeball-photography check ( T_1 ), however as an alternative of dividing our prognosis into two outcomes, let’s divide it into three outcomes:

  • Sufferers are identified as EP-positive, with excessive confidence
  • Sufferers for which the EP check is “ambivalent” and it isn’t doable to tell apart between EP-positive and EP-negative circumstances with excessive confidence
  • Sufferers are identified as EP-negative, with excessive confidence

We take the identical actions within the EP-positive case (admit affected person and start therapy) and the EP-negative case (discharge affected person) as earlier than, however now now we have this center floor. What ought to we do? Properly, we are able to use sources to guage the affected person extra rigorously. Maybe there’s some type of blood check ( T_2 ), which prices $100, however improves our means to tell apart between EP-positive and EP-negative populations. It’s dearer than the $1 check, however a lot inexpensive than the $5000 run of exams we utilized in false optimistic circumstances in our instance.

How can we consider check ( T_2 )?

Bivariate regular distributions

Let’s say that ( T_2 ) additionally has a numeric consequence ( y ) from 0 to 100, and it has a Gaussian distribution as properly, in order that exams ( T_1 ) and ( T_2 ) return a pair of values ( (x,y) ) with a bivariate regular distribution, specifically each ( x ) and ( y ) could be described by their imply values ( mu_x, mu_y ), commonplace deviations ( sigma_x, sigma_y ) and correlation coefficient ( rho ), in order that the covariance matrix ( operatorname{cov}(x,y) = start{bmatrix}S_{xx} & S_{xy} cr S_{xy} & S_{yy}finish{bmatrix} ) could be calculated with ( S_{xx} = sigma_x{}^2, S_{yy} = sigma_y{}^2, S_{xy} = rhosigma_xsigma_y. )

  • When a affected person has EP (situation A), the second-order statistics of ( (x,y) ) could be described as

    • ( mu_x = 55, mu_y = 57 )
    • ( sigma_x=5.3, sigma_y=4.1 )
    • ( rho = 0.91 )

  • When a affected person doesn’t have EP (situation B), the second-order statistics of ( (x,y) ) could be described as

    • ( mu_x = 40, mu_y = 36 )
    • ( sigma_x=5.9, sigma_y=5.2 )
    • ( rho = 0.84 )

The covariance matrix could also be unfamiliar to you, but it surely’s not very sophisticated. (Nonetheless, in the event you don’t like the mathematics, simply skip to the graphs beneath.) Every of the entries of the covariance matrix is merely the anticipated worth of the product of the variables in query after eradicating the imply, so with a pair of zero-mean random variables ( (x’,y’) ) with ( x’ = x – mu_x, y’=y-mu_y ), the covariance matrix is simply ( operatorname{cov}(x’,y’) = start{bmatrix}E[x’^2] & E[x’y’] cr E[x’y’] & E[y’^2] finish{bmatrix} )

With the intention to assist visualize this, let’s graph the 2 situations.

To begin with, we have to know how you can generate pseudorandom values with these distributions. If we generate two unbiased Gaussian random variables ( (u,v) ) with zero imply and unit commonplace deviation, then the covariance matrix is simply ( start{bmatrix}1 & 0 cr 0 & 1end{bmatrix} ). We will create new random variables ( (x,y) ) as a linear mixture of ( u ) and ( v ):

$$start{aligned}x &= a_1u + b_1v cr
y &= a_2u + b_2v

On this case, ( operatorname{cov}(x,y)=start{bmatrix}E[x^2] & E[xy] cr E[xy] & E[y^2] finish{bmatrix}
= start{bmatrix}a_1{}^2 + b_1{}^2 & a_1a_2 + b_1b_2 cr a_1a_2 + b_1b_2 & a_2{}^2 + b_2{}^2 finish{bmatrix}. ) For example for computing this, ( E[x^2] = E[(a_1u+b_1v)^2] = a_1{}^2E[u^2] + 2a_1b_1E[uv] + b_1{}^2E[v^2] = a_1{}^2 + b_1{}^2 ) since ( E[u^2]=E[v^2] = 1 ) and ( E[uv]=0 ).

We will select values ( a_1, a_2, b_1, b_2 ) in order that we obtain the specified covariance matrix:

a_1 &= sigma_x cos theta_x cr
b_1 &= sigma_x sin theta_x cr
a_2 &= sigma_y cos theta_y cr
b_2 &= sigma_y sin theta_y cr

which yields ( operatorname{cov}(x,y)
= start{bmatrix}sigma_x^2 & sigma_xsigma_ycos(theta_x -theta_y) cr sigma_xsigma_ycos(theta_x -theta_y) & sigma_y^2 finish{bmatrix}, ) and subsequently we are able to select any ( theta_x, theta_y ) such that ( cos(theta_x -theta_y) = rho. ) Particularly, we are able to all the time select ( theta_x = 0 ) and ( theta_y = cos^{-1}rho ), in order that

a_1 &= sigma_x cr
b_1 &= 0 cr
a_2 &= sigma_y rho cr
b_2 &= sigma_y sqrt{1-rho^2} cr

and subsequently

x &= mu_x + sigma_x u cr
y &= mu_y + rhosigma_y u + sqrt{1-rho^2}sigma_y v

is a doable methodology of setting up ( (x,y) ) from unbiased unit Gaussian random variables ( (u,v). ) (For the imply values, we simply added them in on the finish.)

OK, so let’s use this to generate samples from the 2 situations A and B, and graph them:

from scipy.stats import chi2
import matplotlib.colours
colorconv = matplotlib.colours.ColorConverter()

Coordinate2D = namedtuple('Coordinate2D','x y')

class Gaussian2D(namedtuple('Gaussian','mu_x mu_y sigma_x sigma_y rho title id coloration')):
    def mu(self):
        """ imply """
        return Coordinate2D(self.mu_x, self.mu_y)
    def cov(self):
        """ covariance matrix """
        crossterm = self.rho*self.sigma_x*self.sigma_y
        return np.array([[self.sigma_x**2, crossterm],
                         [crossterm, self.sigma_y**2]])
    def pattern(self, N, r=np.random):
        """ generate N random samples """
        u = r.randn(N)
        v = r.randn(N)
        return self._transform(u,v)
    def _transform(self, u, v):
        """ rework from IID (u,v) to (x,y) with this distribution """
        rhoc = np.sqrt(1-self.rho**2)
        x = self.mu_x + self.sigma_x*u 
        y = self.mu_y + self.sigma_y*self.rho*u + self.sigma_y*rhoc*v
        return x,y
    def uv2xy(self, u, v):
        return self._transform(u,v)
    def xy2uv(self, x, y):
        rhoc = np.sqrt(1-self.rho**2)
        u = (x-self.mu_x)/self.sigma_x
        v = ((y-self.mu_y) - self.sigma_y*self.rho*u)/rhoc/self.sigma_y
        return u,v
    def contour(self, c, npoint=360):
        generate elliptical contours
        enclosing a fraction c of the inhabitants
        (c could be a vector)
        R^2 is a chi-squared distribution with 2 levels of freedom:
        r = np.sqrt(chi2.ppf(c,2))
        if np.dimension(c) > 1 and len(np.form(c)) == 1:
            r = np.atleast_2d(r).T
        th = np.arange(npoint)*2*np.pi/npoint
        return self._transform(r*np.cos(th), r*np.sin(th))
    def pdf_exponent(self, x, y):
        xdelta = x - self.mu_x
        ydelta = y - self.mu_y
        return -0.5/(1-self.rho**2)*(
            - 2.0*self.rho*xdelta*ydelta/self.sigma_x/self.sigma_y
            + ydelta**2/self.sigma_y**2
    def pdf_scale(self):
        return 1.0/2/np.pi/np.sqrt(1-self.rho**2)/self.sigma_x/self.sigma_y
    def pdf(self, x, y):
        """ likelihood density operate """
        q = self.pdf_exponent(x,y)
        return self.pdf_scale * np.exp(q)
    def logpdf(self, x, y):
        return np.log(self.pdf_scale) + self.pdf_exponent(x,y)
    def logpdf_coefficients(self):
        returns a vector (a,b,c,d,e,f)
        such that log(pdf(x,y)) = ax^2 + bxy + cy^2 + dx + ey + f
        f0 = np.log(self.pdf_scale)
        r = -0.5/(1-self.rho**2)
        a = r/self.sigma_x**2
        b = r*(-2.0*self.rho/self.sigma_x/self.sigma_y)
        c = r/self.sigma_y**2
        d = -2.0*a*self.mu_x -b*self.mu_y
        e = -2.0*c*self.mu_y -b*self.mu_x
        f = f0 + a*self.mu_x**2 + c*self.mu_y**2 + b*self.mu_x*self.mu_y
        return np.array([a,b,c,d,e,f])
    def venture(self, axis):
        Returns a 1-D distribution on the required axis
        if isinstance(axis, basestring):
            if axis == 'x':
                mu = self.mu_x
                sigma = self.sigma_x
            elif axis == 'y':
                mu = self.mu_y
                sigma = self.sigma_y
                elevate ValueError('axis have to be x or y')
            # assume linear mixture of x,y
            a,b = axis
            mu = a*self.mu_x + b*self.mu_y
            sigma = np.sqrt((a*self.sigma_x)**2 
        return Gaussian1D(mu,sigma,self.title,,self.coloration)
    def slice(self, x=None, y=None):
        Returns data (w, mu, sigma) on the likelihood distribution
        with x or y constrained:
        w:     likelihood density throughout all the slice
        mu:    imply worth of the pdf inside the slice
        sigma: commonplace deviation of the pdf inside the slice
        if x is None and y is None:
            elevate ValueError("At the least certainly one of x or y have to be a worth")
        rhoc = np.sqrt(1-self.rho**2)
        if y is None:
            w = scipy.stats.norm.pdf(x, self.mu_x, self.sigma_x)
            mu = self.mu_y + self.rho*self.sigma_y/self.sigma_x*(x-self.mu_x)
            sigma = self.sigma_y*rhoc
            # x is None
            w = scipy.stats.norm.pdf(y, self.mu_y, self.sigma_y)
            mu = self.mu_x + self.rho*self.sigma_x/self.sigma_y*(y-self.mu_y)
            sigma = self.sigma_x*rhoc
        return w, mu, sigma
    def slicefunc(self, which):
        rhoc = np.sqrt(1-self.rho**2)
        if which == 'x':
            sigma = self.sigma_y*rhoc
            a = self.rho*self.sigma_y/self.sigma_x
            def f(x):
                w = scipy.stats.norm.pdf(x, self.mu_x, self.sigma_x)
                mu = self.mu_y + a*(x-self.mu_x)
                return w,mu,sigma
        elif which == 'y':
            sigma = self.sigma_x*rhoc
            a = self.rho*self.sigma_x/self.sigma_y
            def f(y):
                w = scipy.stats.norm.pdf(y, self.mu_y, self.sigma_y)
                mu = self.mu_x + a*(y-self.mu_y)
                return w,mu,sigma
            elevate ValueError("'which' have to be x or y")
        return f
N = 100000

distA = Gaussian2D(mu_x=55,mu_y=57,sigma_x=5.3,sigma_y=4.1,rho=0.91,title="A (EP-positive)",id='A',coloration="purple")
distB = Gaussian2D(mu_x=40,mu_y=36,sigma_x=5.9,sigma_y=5.2,rho=0.84,title="B (EP-negative)",id='B',coloration="#8888ff")
xA,yA = distA.pattern(N)
xB,yB = distB.pattern(N)

def scatter_samples(ax,xyd_list,contour_list=(),**kwargs):
    Kmute = 1 if not contour_list else 0.5
    for x,y,dist in xyd_list:
        mutedcolor = colorconv.to_rgb(dist.coloration)
        mutedcolor = [c*Kmute+(1-Kmute) for c in mutedcolor]
        if not contour_list:
    for x,y,dist in xyd_list:
        # Now draw contours for sure chances
        th = np.arange(1200)/1200.0*2*np.pi
        u = np.cos(th)
        v = np.sin(th)
        first = True
        for p in contour_list:
            cx,cy = dist.contour(p)
            kwargs = {}
            if first:
                first = False

    ax.legend(loc="decrease proper",markerscale=10)
    title="Scatter pattern plot"
    if contour_list:
        title += (', %s%% CDF ellipsoid contours proven' 
                  % (', '.be a part of('%.0f' % (p*100) for p in contour_list)))
    ax.set_title(title, fontsize=10)
for x,y,desc in [(xA, yA,'A'),(xB,yB,'B')]:
    print "Covariance matrix for case %s:" % desc
    C = np.cov(x,y)
    print C
    sx = np.sqrt(C[0,0])
    sy = np.sqrt(C[1,1])
    rho = C[0,1]/sx/sy
    print "pattern sigma_x = %.3f" % sx 
    print "pattern sigma_y = %.3f" % sy
    print "pattern rho = %.3f" % rho
Covariance matrix for case A:
[[ 28.06613839  19.74199382]
 [ 19.74199382  16.76597717]]
pattern sigma_x = 5.298
pattern sigma_y = 4.095
pattern rho = 0.910
Covariance matrix for case B:
[[ 34.6168817   25.69386711]
 [ 25.69386711  27.05651845]]
pattern sigma_x = 5.884
pattern sigma_y = 5.202
pattern rho = 0.840

It could appear unusual, however having the outcomes of two exams (( x ) from check ( T_1 ) and ( y ) from check ( T_2 )) provides extra helpful data than the results of every check thought-about by itself. We’ll come again to this concept a bit bit later.

The extra quick query is: given the pair of outcomes ( (x,y) ), how would we determine whether or not the affected person has ( EP ) or not? With simply check ( T_1 ) we might merely declare an EP-positive prognosis if ( x > x_0 ) for some threshold ( x_0 ). With two variables, some type of inequality is concerned, however how can we determine?

Bayes’ Rule

We’re enormously indebted to the varied European heads of state and faith throughout a lot of the 18th century (the Age of Enlightenment) for merely leaving individuals alone. (OK, this wasn’t universally true, however lots of the monarchies turned a blind eye in the direction of intellectualism.) This lack of interference and oppression resulted in quite a few mathematical and scientific discoveries, certainly one of which was Bayes’ Rule, named after Thomas Bayes, a British clergyman and mathematician. Bayes’ Rule was revealed posthumously in An Essay in the direction of fixing a Downside within the Doctrine of Probabilities, and later inflicted on throngs of undergraduate college students of likelihood and statistics.

The essential thought includes conditional chances and jogs my memory of the logical converse. As a hypothetical instance, suppose we all know that 95% of Dairy Queen prospects are from the USA and that 45% of these US residents who go to Dairy Queen like peppermint ice cream, whereas 72% of non-US residents like peppermint ice cream. We’re in line to get some ice cream, and we discover that the particular person in entrance of us orders peppermint ice cream. Can we make any prediction of the likelihood that this particular person is from the US?

Bayes’ Rule relates these two situations. Let ( A ) characterize the situation {that a} Dairy Queen buyer is a US resident, and ( B ) characterize that they like peppermint ice cream. Then ( P(A | B) = frac A)P(A){P(B)} ), which is de facto simply an algebraic rearrangement of the expression of the joint likelihood that ( A ) and ( B ) are each true: ( P(AB) = P(A | B)P(B) = P(B | A)P(A) ). Utilized to our Dairy Queen instance, now we have ( P(A) = 0.95 ) (95% of Dairy Queen prospects are from the US) and ( P(B | A) = 0.45 ) (45% of consumers like peppermint ice cream, on condition that they’re from the US). However what’s ( P(B) ), the likelihood {that a} Dairy Queen buyer likes peppermint ice cream? Properly, it’s the sum of the all of the constituent joint chances the place the client likes peppermint ice cream. For instance, ( P(AB) = P(B | A)P(A) = 0.45 occasions 0.95 = 0.4275 ) is the joint likelihood {that a} Dairy Queen buyer is from the US and likes peppermint ice cream, and ( P(bar{A}B) = P(B | bar{A})P(bar{A}) = 0.72 occasions 0.05 = 0.036 ) is the joint likelihood {that a} Dairy Queen buyer is not from the US and likes peppermint ice cream. Then ( P(B) = P(AB) + P(bar{A}B) = 0.4275 + 0.036 = 0.4635 ). (46.35% of all Dairy Queen prospects like peppermint ice cream.) The ultimate utility of Bayes’ Rule tells us

$$P(A | B) = frac A)P(A){P(B)} = frac{0.45 occasions 0.95}{0.4635} approx 0.9223$$

and subsequently if we see somebody order peppermint ice cream at Dairy Queen, there’s a whopping 92.23% probability they aren’t from the US.

Let’s return to our embolary pulmonism situation the place an individual takes each exams ( T_1 ) and ( T_2 ), with outcomes ( R = (x=45, y=52) ). Can we estimate the likelihood that this particular person has EP?

N = 500000
xA,yA = distA.pattern(N)
xB,yB = distB.pattern(N)


We definitely aren’t going to have the ability to discover the reply precisely simply from taking a look at this chart, but it surely seems like an nearly sure case of A being true — that’s, ( R: x=45, y=52 ) implies that the affected person in all probability has EP. Let’s determine it out as

$$P(A | R) = frac A)P(A){P(R)}.$$

Bear in mind we mentioned earlier that the bottom price, which is the likelihood of any given particular person presenting signs having EP earlier than any testing, is ( P(A) = 40times 10^{-6} ). (This is called the a priori likelihood, each time this Bayesian stuff will get concerned.) The opposite two chances ( P(R | A) ) and ( P(R) ) are technically infinitesimal, as a result of they’re a part of steady likelihood distributions, however we are able to handwave and say that ( R ) is de facto the situation that the outcomes are ( 45 le x le 45 + dx ) and ( 52 le y le 52+dy ) for some infinitesimal interval widths ( dx, dy ), through which case ( P(R | A) = p_A(R),dx,dy ) and ( P(R) = P(R | A)P(A) + P(R | B)P(B) = p_A(R)P(A),dx,dy + p_B(R)P(B),dx,dy ) the place ( p_A ) and ( p_B ) are the likelihood density features. Substituting this all in we get

$$P(A | R) = frac{p_A(R)P(A)}{p_A(R)P(A)+p_B(R)P(B)}$$

The type of the bivariate regular distribution is just not too sophisticated, only a bit unwieldy:

$$p(x,y) = frac{1}{2pisqrt{1-rho^2}sigma_xsigma_y}e^{q(x,y)}$$


$$q(x,y) = -frac{1}{2(1-rho^2)}left(frac{(x-mu_x)^2}{sigma_x{}^2}-2rhofrac{(x-mu_x)(y-mu_y)}{sigma_xsigma_y}+frac{(y-mu_y)^2}{sigma_y{}^2}proper)$$

and the remainder is simply quantity crunching:

PA_total = 40e-6
pA = distA.pdf(x1, y1)
pB = distB.pdf(x1, y1)
print "pA(%.1f,%.1f) = %.5g" % (x1,y1,pA)
print "pB(%.1f,%.1f) = %.5g" % (x1,y1,pB)
print ("Bayes' rule consequence: p(A | x=%.1f, y=%.1f) = %.5g" % 
pA(45.0,52.0) = 0.0014503
pB(45.0,52.0) = 4.9983e-07
Bayes' rule consequence: p(A | x=45.0, y=52.0) = 0.104

Wow, that’s counterintuitive. This consequence worth ( R ) lies a lot nearer to the likelihood “cloud” of A = EP-positive than B = EP-negative, however Bayes’ Rule tells us there’s solely a couple of 10.4% likelihood {that a} affected person with check outcomes ( (x=45, y=52) ) has EP. And it’s due to the very low incidence of EP.

There’s something we are able to do to make studying the graph helpful, and that’s to plot a parameter I’m going to name ( lambda(x,y) ), which is the logarithm of the ratio of likelihood densities:

$$lambda(x,y) = ln frac{p_A(x,y)}{p_B(x,y)} = ln p_A(x,y) – ln p_B(x,y)$$

Truly, we’ll plot ( lambda_{10}(x,y) = lambda(x,y) / ln 10 = log_{10} frac{p_A(x,y)}{p_B(x,y)} ).

This parameter is helpful as a result of Bayes’ rule calculates

P(A | x,y) &= frac{p_A(x,y) P_A} { p_A(x,y) P_A + p_B(x,y) P_B} cr
&= frac{p_A(x,y)/p_B(x,y) P_A} {p_A(x,y)/p_B(x,y) P_A + P_B} cr
&= frac{e^{lambda(x,y)} P_A} {e^{lambda(x,y)} P_A + P_B} cr
&= frac{1}{1 + e^{-lambda(x,y)}P_B / P_A}

and for any desired ( P(A | x,y) ) we are able to work out the equal worth of ( lambda(x,y) = – ln left(frac{P_A}{P_B}((1/P(A | x,y) – 1)proper). )

Which means for a hard and fast worth of ( lambda ), then ( P(A | lambda) = frac{1}{1 + e^{-lambda}P_B / P_A}. )

For bivariate Gaussian distributions, the ( lambda ) parameter can be helpful as a result of it’s a quadratic operate of ( x ) and ( y ), so curves of fixed ( lambda ) are conic sections (strains, ellipses, hyperbolas, or parabolas).

def jcontourp(ax,x,y,z,ranges,majorfunc,coloration=None,fmt=None, **kwargs):
    linewidths = [1 if majorfunc(l) else 0.4 for l in levels]
    cs = ax.contour(x,y,z,ranges,
    labeled = [l for l in cs.levels if majorfunc(l)]
    ax.clabel(cs, labeled, inline=True, fmt="%s", fontsize=10)
xv = np.arange(10,80.01,0.1)
yv = np.arange(10,80.01,0.1)
x,y = np.meshgrid(xv,yv)
def lambda10(distA,distB,x,y):
    return (distA.logpdf(x, y)-distB.logpdf(x, y))/np.log(10)

fig = plt.determine()
ax = fig.add_subplot(1,1,1)
print "lambda10(x=%.1f,y=%.1f) = %.2f" % (x1,y1,lambda10(distA,distB,x1,y1))
ranges = np.union1d(np.arange(-10,10), np.arange(-200,100,10))
def levelmajorfunc(degree):
    if -10 <= degree <= 10:
        return int(degree) % 5 == 0
        return int(degree) % 25 == 0
ax.set_xlim(xv.min(), xv.max()+0.001)
ax.set_ylim(yv.min(), yv.max()+0.001)
ax.set_title('Scatter pattern plot with contours = $lambda_{10}$ values');
lambda10(x=45.0,y=52.0) = 3.46

We’ll choose one explicit ( lambda_{10} ) worth as a threshold ( L_{10} = L/ ln 10 ), and if ( lambda_{10} > L_{10} ) then we’ll declare situation ( a ) (the affected person is identified as EP-positive), in any other case we’ll declare situation ( b ) (the affected person is identified as EP-negative). The only option of ( L_{10} ) is the one which maximizes anticipated worth.

Bear in mind how we did this within the case with solely the one check ( T_1 ) with consequence ( x ): we selected threshold ( x_0 ) primarily based on the purpose the place ( {partial E over partial x_0} = 0 ); in different phrases, a change in prognosis didn’t change the anticipated worth at this level. We will do the identical factor right here:

$$0 = {partial E[v] over partial {L}} = P(A | lambda=L)(v_{Ab}-v_{Aa}) + P(B | lambda=L)(v_{Bb}-v_{Ba})$$

With our earlier definitions

$$rho_v = -frac{v_{Bb}-v_{Ba}}{v_{Ab}-v_{Aa}}, qquad rho_p = P_B / P_A$$

then the equation for ( L ) turns into ( 0 = P(A | lambda=L) – rho_v P(B | lambda=L) = P(A | lambda=L) – rho_v (1-P(A | lambda=L)) ), which simplifies to

$$P(A | lambda=L) = frac{rho_v}{rho_v+1} = frac{1}{1+1/rho_v} .$$

However we already outlined ( lambda ) because the group of factors ( (x,y) ) which have equal likelihood ( P(A | lambda) = frac{1}{1 + e^{-lambda}P_B / P_A} = frac{1}{1 + rho_p e^{-lambda}} ) so

$$frac{1}{1+1/rho_v} = frac{1}{1 + rho_p e^{-L}}$$

which happens when ( L = ln rho_v rho_p. )

In our EP instance,

rho_p &= p_B/p_A = 0.99996 / 0.00004 = 24999 cr
rho_v &= -frac{v_{Bb}-v_{Ba}}{v_{Ab}-v_{Aa}} = -5000 / -9.9times 10^6 approx 0.00050505 cr
rho_vrho_p &approx 12.6258 cr
L &= ln rho_vrho_p approx 2.5357 cr
L_{10} &= log_{10}rho_vrho_p approx 1.1013

and we are able to full the evaluation empirically by wanting on the fraction of pseudorandomly-generated pattern factors the place ( lambda_{10} < L_{10} ); that is an instance of Monte Carlo evaluation.

class Quadratic1D(namedtuple('Quadratic1D','a b c')):
    Q(x) = a*x*x + b*x + c
    __slots__ = ()
    def x0(self):
        return -self.b/(2.0*self.a)
    def q0(self):
        return self.c - self.a*self.x0**2
    def __call__(self,x):
        return self.a*x*x + self.b*x + self.c
    def clear up(self, q):
        D = self.b*self.b - 4*self.a*(self.c-q)
        sqrtD = np.sqrt(D)
        return np.array([-self.b-sqrtD, -self.b+sqrtD])/(2*self.a)

class QuadraticLissajous(namedtuple('QuadraticLissajous','x0 y0 Rx Ry phi')):
    A parametric curve as a operate of theta:
        x = x0 + Rx * cos(theta)
        y = y0 + Ry * sin(theta+phi)
    __slots__ = ()
    def __call__(self, theta):
        return (self.x0 + self.Rx * np.cos(theta),
                self.y0 + self.Ry * np.sin(theta+self.phi))
class Quadratic2D(namedtuple('Quadratic2D','a b c d e f')):
    Bivariate quadratic operate 
    Q(x,y) = a*x*x + b*x*y + c*y*y + d*x + e*y + f
           = a*(x-x0)*(x-x0) + b*(x-x0)*(y-y0) + c*(y-y0)*(y-y0) + q0
           = s*(u*u + v*v) + q0 the place s = +/-1
    (Warning: this implementation assumes convexity,
    that's, b*b < 4*a*c, so hyperboloids/paraboloids should not dealt with.)
    __slots__ = ()
    def discriminant(self):
        return self.b**2 - 4*self.a*self.c
    def x0(self):
        return (2*self.c*self.d - self.b*self.e)/self.discriminant
    def y0(self):
        return (2*self.a*self.e - self.b*self.d)/self.discriminant
    def q0(self):
        x0 = self.x0
        y0 = self.y0
        return self.f - self.a*x0*x0 - self.b*x0*y0 - self.c*y0*y0
    def _Kcomponents(self):
        s = 1 if self.a > 0 else -1
        r = s*self.b/2.0/np.sqrt(self.a*self.c)
        rc = np.sqrt(1-r*r)
        Kux = rc*np.sqrt(self.a*s)
        Kvx = r*np.sqrt(self.a*s)
        Kvy = np.sqrt(self.c*s)
        return Kux, Kvx, Kvy
    def Kxy2uv(self):
        Kux, Kvx, Kvy = self._Kcomponents()
        return np.array([[Kux, 0],[Kvx, Kvy]])
    def Kuv2xy(self):
        Kux, Kvx, Kvy = self._Kcomponents()
        return np.array([[1.0/Kux, 0],[-1.0*Kvx/Kux/Kvy, 1.0/Kvy]])        
    def transform_xy2uv(self):
        Kxy2uv = self.Kxy2uv
        x0 = self.x0
        y0 = self.y0
        def rework(x,y):
            return, [x-x0,y-y0])
        return rework
    def transform_uv2xy(self):
        Kuv2xy = self.Kuv2xy
        x0 = self.x0
        y0 = self.y0
        def rework(u,v):
            return, [u,v]) + [[x0],[y0]]
        return rework
    def uv_radius(self, q):
        Returns R such that options (u,v) of Q(x,y) = q
        lie inside the vary [-R, R], 
        or None if there are not any options.
        s = 1 if self.a > 0 else -1
        D = (q-self.q0)*s
        return np.sqrt(D) if D >= 0 else None
    def _xy_radius_helper(self, q, z):
        D = (self.q0 - q) * 4 * z / self.discriminant
        if D < 0:
            return None
            return np.sqrt(D)
    def x_radius(self, q):
        Returns Rx such that options (x,y) of Q(x,y) = q
        lie inside the vary [x0-Rx, x0+Rx], 
        or None if there are not any options.
        Equal to uv_radius(q)/Kux, however we are able to clear up extra instantly
        return self._xy_radius_helper(q, self.c)
    def y_radius(self, q):
        Returns Rx such that options (x,y) of Q(x,y) = q
        lie inside the vary [x0-Rx, x0+Rx], 
        or None if there are not any options.
        Equal to uv_radius(q)/Kux, however we are able to clear up extra instantly
        return self._xy_radius_helper(q, self.a)
    def lissajous(self, q):
        Returns a QuadraticLissajous with x0, y0, Rx, Ry, phi such that 
        the options (x,y) of Q(x,y) = q
        could be written:
          x = x0 + Rx * cos(theta)
          y = y0 + Ry * sin(theta+phi)
        Rx and Ry and phi could every return None if no such resolution exists.  
        D = self.discriminant
        x0 = (2*self.c*self.d - self.b*self.e)/D
        y0 = (2*self.a*self.e - self.b*self.d)/D
        q0 = self.f - self.a*x0*x0 - self.b*x0*y0 - self.c*y0*y0
        Dx = 4 * (q0-q) * self.c / D
        Rx = None if Dx < 0 else np.sqrt(Dx)
        Dy = 4 * (q0-q) * self.a / D
        Ry = None if Dy < 0 else np.sqrt(Dy)
        phi = None if D > 0 else np.arcsin(self.b / (2*np.sqrt(self.a*self.c)))
        return QuadraticLissajous(x0,y0,Rx,Ry,phi)
    def contour(self, q, npoints=360):
        Returns a pair of arrays x,y such that
        Q(x,y) = q
        s = 1 if self.a > 0 else -1
        R = np.sqrt((q-self.q0)*s)
        th = np.arange(npoints)*2*np.pi/npoints
        u = R*np.cos(th)
        v = R*np.sin(th)
        return self.transform_uv2xy(u,v)
    def constrain(self, x=None, y=None):
        if x is None and y is None:
            return self
        if x is None:
            # return a operate in x
            return Quadratic1D(self.a,
                               self.d + y*self.b,
                               self.f + y*self.e + y*y*self.c)
        if y is None:
            # return a operate in y
            return Quadratic1D(self.c,
                               self.e + x*self.b,
                               self.f + x*self.d + x*x*self.a)
        return self(x,y)
    def __call__(self, x, y):
        return (self.a*x*x 

def decide_limits(*args, **kwargs):
    s = kwargs.get('s', 6)
    xmin = None
    xmax = None
    for xbatch in args:
        xminb = min(xbatch)
        xmaxb = max(xbatch)
        mu = np.imply(xbatch)
        std = np.std(xbatch)
        xminb = min(mu-s*std, xminb)
        xmaxb = max(mu+s*std, xmaxb)
        if xmin is None:
            xmin = xminb
            xmax = xmaxb
            xmin = min(xmin,xminb)
            xmax = max(xmax,xmaxb)
    # Quantization
    q = kwargs.get('q')
    if q is just not None:
        xmin = np.ground(xmin/q)*q
        xmax = np.ceil(xmax/q)*q
    return xmin, xmax
def separation_plot(xydistA, xydistB, Q, L, ax=None, xlim=None, ylim=None):
    L10 = L/np.log(10)
    if ax is None:
        fig = plt.determine()
        ax = fig.add_subplot(1,1,1)
    xA,yA,distA = xydistA
    xB,yB,distB = xydistB

    xc,yc = Q.contour(L)
        coloration="inexperienced", linewidth=1.5, dashes=[5,2],
        label="$lambda_{10} = %.4f$" % L10)
    ax.legend(loc="decrease proper",markerscale=10, 
    if xlim is None:
        xlim = decide_limits(xA,xB,s=6,q=10)
    if ylim is None:
        ylim = decide_limits(yA,yB,s=6,q=10)
    xv = np.arange(xlim[0],xlim[1],0.1)
    yv = np.arange(ylim[0],ylim[1],0.1)
    x,y = np.meshgrid(xv,yv)

    ax.set_title('Scatter pattern plot with contours = $lambda_{10}$ values')

def separation_report(xydistA, xydistB, Q, L):
    L10 = L/np.log(10)
    for x,y,dist in [xydistA, xydistB]:
        print "Separation of samples in %s by L10=%.4f" % (,L10)
        lam = Q(x,y)
        lam10 = lam/np.log(10)
        print "  Vary of lambda10: %.4f to %.4f" % (np.min(lam10), np.max(lam10))
        n = np.dimension(lam)
        p = np.count_nonzero(lam < L) * 1.0 / n
        print "  lambda10 <  L10: %.5f" % p
        print "  lambda10 >= L10: %.5f" % (1-p)
L = np.log(5000 / 9.9e6 * 24999)
C = distA.logpdf_coefficients - distB.logpdf_coefficients
Q = Quadratic2D(*C)
separation_plot((xA,yA,distA),(xB,yB,distB), Q, L)
separation_report((xA,yA,distA),(xB,yB,distB), Q, L)
Separation of samples in A by L10=1.1013
  Vary of lambda10: -3.4263 to eight.1128
  lambda10 <  L10: 0.01760
  lambda10 >= L10: 0.98240
Separation of samples in B by L10=1.1013
  Vary of lambda10: -36.8975 to 4.0556
  lambda10 <  L10: 0.99834
  lambda10 >= L10: 0.00166

Or we are able to decide the outcomes by numerical integration of likelihood density.

The maths beneath isn’t troublesome, simply tedious; for every of the 2 Gaussian distributions for A and B, I chosen a sequence of 5000 intervals (10001 x-axis factors) from ( mu_x-8sigma_x ) to ( mu_x + 8sigma_x ), and used Simpson’s Rule to combine the likelihood density ( f_x(x_i) ) at every level ( x_i ), on condition that

$$x approx x_i quadRightarrowquad f_x(x) approx f_x(x_i) = p_x(x_i) left(F_{x_i}(y_{i2}) – F_{x_i}(y_{i1})proper)$$

the place

  • ( p_x(x_i) ) is the likelihood density that ( x=x_i ) with ( y ) unspecified
  • ( F_{x_i}(y_0) ) is the 1-D cumulative distribution operate, given ( x=x_i ), that ( y<y_0 )
  • ( y_{i1} ) and ( y_{i2} ) are both

    • the 2 options of ( lambda(x_i,y) = L )
    • or zero, if there are not any options or one resolution

and the pattern factors ( x_i ) are positioned extra intently collectively nearer to the extremes of the contour ( lambda(x,y)=L ) to seize the suddenness of the change.

Anyway, right here’s the consequence, which is properly according to the Monte Carlo evaluation:

def assert_ordered(*args, **kwargs):
    if len(args) < 2:

    rthresh = kwargs.get('rthresh',1e-10)
    label = kwargs.get('label',None)
    label="" if label is None else label+': '
    xmin = args[0]
    xmax = args[-1]
    if len(args) == 2:
        # not very fascinating case
        xthresh = rthresh*(xmax+xmin)/2.0
        xthresh = rthresh*(xmax-xmin)
    xprev = xmin
    for x in args[1:]:
        assert x - xprev >= -xthresh, "%spercents > %s + %g" % (label,xprev,x,xthresh)
        xprev = x

def arccos_sat(x):
    if x <= -1:
        return np.pi
    if x >= 1:
        return 0
    return np.arccos(x)

def simpsons_rule_points(xlist, bisect=True):
    Generator for Simpson's rule
    xlist:  arbitrary factors in rising order
    bisect: whether or not or to not add bisection factors
    returns a generator of weights w (if bisect=False) 
      or tuples (w,x) (if bisect = True)
    such that the integral of f(x) dx over the listing of factors xlist
    is roughly equal to:
    sum(w*f(x) for w,x in simpsons_rule_points(xlist))
    The values of x returned are x[i], (x[i]+x[i+1])/2, x[i+1]
    with relative weights dx/6, 4*dx/6, dx/6 for every interval
    [x[i], x[i+1]]
    xiter = iter(xlist)
    xprev = xiter.subsequent()
    w2 = 0
    x2 = None
    if bisect:
        for x2 in xiter:
            x0 = xprev
            dx = x2-x0
            x1 = x0 + dx/2.0
            xprev = x2
            w6 = dx/6.0
            w0 = w2 + w6
            yield (w0, x0)
            w1 = 4*w6
            yield (w1, x1)
            w2 = w6
        if x2 is just not None:
            yield (w2, x2)
        for x1 in xiter:
            x0 = xprev
                x2 = xiter.subsequent()
            besides StopIteration:
                elevate ValueError("Should have an odd variety of factors")
            dx = x2-x0
            xprev = x2
            w6 = dx/6.0
            w0 = w2 + w6
            yield w0
            w1 = 4*w6
            yield w1
            w2 = w6
        if x2 is just not None:
            yield w2
def estimate_separation_numerical(dist, Q, L, xmin, xmax, Nintervals=5000,
                                  return_pair=False, sampling_points=None):
    Numerical estimate of every row of confusion matrix
    utilizing N integration intervals (every bisected to make use of Simpson's Rule)
    protecting the interval [xmin,xmax].
    dist:        Gaussian2D distribution
    Q:           Quadratic2D operate Q(x,y)
    L:           lambda threshold: 
                   diagnose as A when Q(x,y) > L,
                   in any other case as B
    xmin:       begin x-coordinate
    xmax:        finish x-coordinate
    Nintervals:  variety of integration intervals
    Returns p, the place p is the likelihood of Q(x,y) > L for that
    If return_pair is true, returns [p,q];
    if [x1,x2] covers a lot of the distribution, then p+q must be near 1.
    Qliss = Q.lissajous(L)
    xqr = Qliss.Rx
    xqmu = Qliss.x0
    # Decide vary of options:
    #    no options if xqr is None,
    #    in any other case Q(x,y) > L if x in Rx = [xqmu-xqr, xqmu+xqr]
    # Cowl the trivial circumstances first:
    if xqr is None:
        return (0,1) if return_pair else 0
    if (xmax < xqmu - xqr) or (xmin > xqmu + xqr):
        # All options are greater than Nsigma from the imply
        # isigma_left  = (xqmu - xqr - dist.mu_x)/dist.sigma_x
        # isigma_right = (xqmu + xqr - dist.mu_x)/dist.sigma_x
        # print "sigma", isigma_left, isigma_right
        return (0,1) if return_pair else 0
    # We need to cowl the vary xmin, xmax.
    # Improve protection close to the ends of the lambda threshold,
    # xqmu +/- xqr the place the habits adjustments extra quickly,
    # through the use of factors at -cos(theta) inside the resolution house for Q(x,y)>L
    th1 = arccos_sat(-(xmin-xqmu)/xqr)
    th2 = arccos_sat(-(xmax-xqmu)/xqr)
    # print np.array([th1,th2])*180/np.pi, xmin, xqmu-np.cos(th1)*xqr, xqmu-np.cos(th2)*xqr, xmax, (xqmu-xqr, xqmu+xqr)
    assert_ordered(xmin, xqmu-np.cos(th1)*xqr, xqmu-np.cos(th2)*xqr, xmax)
    x_arc_coverage = xqr*(np.cos(th1)-np.cos(th2))   # size alongside x
    x_arc_length = xqr*(th2-th1)                     # size alongside arc
    x_total_length = (xmax-xmin) - x_arc_coverage + x_arc_length
    x1 = xqmu-xqr*np.cos(th1)
    x2 = x1 + x_arc_length
    n = (Nintervals*2) + 1
    xlin = np.linspace(0,x_total_length, n) + xmin
    x = xlin[:]
    y = np.zeros((2,n))
    # Factors alongside arc:
    tol = 1e-10
    i12 = (xlin >= x1 - tol) & (xlin <= x2 + tol)
    angles = th1 + (xlin[i12]-x1)/x_arc_length*(th2-th1)
    x[i12], y[0, i12] = Qliss(np.pi + angles)
    _,      y[1, i12] = Qliss(np.pi - angles)
    x[xlin >= x2] += (x_arc_coverage-x_arc_length)
    fx = dist.slicefunc('x')
    p = 0
    q = 0
    for i,wdx in enumerate(simpsons_rule_points(x, bisect=False)):
        w, mu, sigma = fx(x[i])
        y1 = y[0,i]
        y2 = y[1,i]
        cdf1 = scipy.stats.norm.cdf(y1, mu, sigma)
        cdf2 = scipy.stats.norm.cdf(y2, mu, sigma)
        deltacdf = cdf2-cdf1
        p += wdx*w*deltacdf
        q += wdx*w*(1-deltacdf)
    return (p,q) if return_pair else p 

def compute_confusion_matrix(distA, distB, Q, L, Nintervals=5000, verbose=False):
    C = np.zeros((2,2))
    for i,dist in enumerate([distA,distB]):
        Nsigma = 8
        xmin = dist.mu_x - Nsigma*dist.sigma_x
        xmax = dist.mu_x + Nsigma*dist.sigma_x
        p,q = estimate_separation_numerical(dist, Q, L, xmin, xmax, Nintervals=Nintervals, 
        C[i,:] = [p,q]
        print "%s: p=%g, q=%g, p+q-1=%+g" % (dist.title, p,q,p+q-1)
    return C

confusion_matrix = compute_confusion_matrix(distA, distB, Q, L, verbose=True)
separation_report((xA,yA,distA),(xB,yB,distB), Q, L)
value_matrix = np.array([[0,-5000],[-1e7, -1e5]]) - 100
# Identical worth matrix as the unique one
# (with vAb = $10 million) earlier than however we add $100 value for check T2

C = confusion_matrix[::-1,::-1]
# flip the confusion matrix left-right and top-bottom for B first

def gather_info(C,V,PA,**kwargs):
    information = dict(kwargs)
    C = np.array(C)
    V = np.array(V)
    information['C'] = C
    information['V'] = V
    information['J'] = C*[[1-PA],[PA]]
    return information    

show(show_binary_matrix(C, 'L_{10}=%.4f' % (L/np.log(10)), [distB, distA], 'ba', 40e-6,value_matrix,
info27 = gather_info(C,value_matrix,40e-6,L=L)
A (EP-positive): p=0.982467, q=0.0175334, p+q-1=-8.90805e-09
B (EP-negative): p=0.00163396, q=0.998366, p+q-1=+1.97626e-07
Separation of samples in A by L10=1.1013
  Vary of lambda10: -3.4263 to eight.1128
  lambda10 <  L10: 0.01760
  lambda10 >= L10: 0.98240
Separation of samples in B by L10=1.1013
  Vary of lambda10: -36.8975 to 4.0556
  lambda10 <  L10: 0.99834
  lambda10 >= L10: 0.00166

Report for threshold (L_{10}=1.1013 rightarrow E[v]=)$-119.11

b a
B (EP-negative) 0.99837
A (EP-positive) 0.01753

There! Now we are able to examine that now we have a neighborhood most, by making an attempt barely decrease and better thresholds:

value_matrix = np.array([[0,-5000],[-1e7, -1e5]]) - 100
delta_L = 0.1*np.log(10)
for Li in [L-delta_L, L+delta_L]:
    confusion_matrix = compute_confusion_matrix(distA, distB, Q, Li, verbose=True)
    separation_report((xA,yA,distA),(xB,yB,distB), Q, Li)
    # Identical worth matrix earlier than however we add $100 value for check T2

    C = confusion_matrix[::-1,::-1]
    # flip the confusion matrix left-right and top-bottom for B first

    show(show_binary_matrix(C, 'L_{10}=%.4f' % (Li/np.log(10)),
                               [distB, distA], 'ba', 40e-6,value_matrix, 
A (EP-positive): p=0.984803, q=0.0151974, p+q-1=-8.85844e-09
B (EP-negative): p=0.0018415, q=0.998159, p+q-1=+2.55256e-07
Separation of samples in A by L10=1.0013
  Vary of lambda10: -3.4263 to eight.1128
  lambda10 <  L10: 0.01550
  lambda10 >= L10: 0.98450
Separation of samples in B by L10=1.0013
  Vary of lambda10: -36.8975 to 4.0556
  lambda10 <  L10: 0.99819
  lambda10 >= L10: 0.00181

Report for threshold (L_{10}=1.0013 rightarrow E[v]=)$-119.23

b a
B (EP-negative) 0.99816
A (EP-positive) 0.01520
A (EP-positive): p=0.979808, q=0.0201915, p+q-1=-9.01419e-09
B (EP-negative): p=0.00144637, q=0.998554, p+q-1=+2.1961e-08
Separation of samples in A by L10=1.2013
  Vary of lambda10: -3.4263 to eight.1128
  lambda10 <  L10: 0.02011
  lambda10 >= L10: 0.97989
Separation of samples in B by L10=1.2013
  Vary of lambda10: -36.8975 to 4.0556
  lambda10 <  L10: 0.99850
  lambda10 >= L10: 0.00150

Report for threshold (L_{10}=1.2013 rightarrow E[v]=)$-119.23

b a
B (EP-negative) 0.99855
A (EP-positive) 0.02019

Nice! The sensitivity of threshold appears fairly flat; ( E[v] ) differs by about 12 cents if we alter ( L_{10} = 1.1013 ) to ( L_{10} = 1.0013 ) or ( L_{10} = 1.2013 ). This provides us a bit wiggle room in the long run to shift prices between the false-negative and false-positive circumstances, with out altering the general anticipated value very a lot.

Most notably, although, we’ve decreased the entire value by about $93 through the use of the pair of exams ( T_1, T_2 ) in comparison with the check with simply ( T_1 ). It’s because we shift value from the Ab (false damaging for EP-positive) and Ba (false optimistic for EP-negative) circumstances to the Bb (right for EP-negative) case — everybody has to pay $100 extra, however the possibilities of false positives and false negatives have been enormously decreased.

Don’t keep in mind the statistics from the one-test case? Right here they’re once more:

x0 = 53.1626
C = analyze_binary(dneg, dpos, x0)
value_matrix_T1 = [[0,-5000],[-1e7, -1e5]]
show(show_binary_matrix(C, x0, [dneg, dpos], 'ba', 40e-6,value_matrix_T1,
info17 = gather_info(C,value_matrix_T1,40e-6,x0=x0)

Report for threshold (x_0 = 53.1626 rightarrow E[v]=)$-212.52

b a
B (EP-negative) 0.98716
A (EP-positive) 0.36442

And the equal circumstances for ( v_{Ab}= )$100 million:

Pa_total = 40e-6
x0 = 48.1458
C1 = analyze_binary(dneg, dpos, x0)
value_matrix_T1 = np.array([[0,-5000],[-1e8, -1e5]])
show(show_binary_matrix(C1, x0, [dneg, dpos], 'ba', Pa_total, value_matrix_T1,
info18 = gather_info(C1,value_matrix_T1,Pa_total,x0=x0)

def compute_optimal_L(value_matrix,Pa_total):
    v = value_matrix
    return np.log(-(v[0,0]-v[0,1])/(v[1,0]-v[1,1])*(1-Pa_total)/Pa_total)

value_matrix_T2 = value_matrix_T1 - 100
L = compute_optimal_L(value_matrix_T2, Pa_total)
confusion_matrix = compute_confusion_matrix(distA, distB, Q, L, verbose=True)
separation_report((xA,yA,distA),(xB,yB,distB), Q, L)
# Identical worth matrix earlier than however we add $100 value for check T2

C2 = confusion_matrix[::-1,::-1]
# flip the confusion matrix left-right and top-bottom for B first

show(show_binary_matrix(C2, 'L_{10}=%.4f' % (L/np.log(10)), [distB, distA],
                           'ba', Pa_total,value_matrix_T2,
separation_plot((xA,yA,distA),(xB,yB,distB), Q, L)
info28 = gather_info(C2,value_matrix_T2,Pa_total,L=L)

Report for threshold (x_0 = 48.1458 rightarrow E[v]=)$-813.91

b a
B (EP-negative) 0.91631
A (EP-positive) 0.09796
A (EP-positive): p=0.996138, q=0.00386181, p+q-1=-8.37725e-09
B (EP-negative): p=0.00491854, q=0.995081, p+q-1=+4.32711e-08
Separation of samples in A by L10=0.0973
  Vary of lambda10: -3.4263 to eight.1128
  lambda10 <  L10: 0.00389
  lambda10 >= L10: 0.99611
Separation of samples in B by L10=0.0973
  Vary of lambda10: -36.8975 to 4.0556
  lambda10 <  L10: 0.99514
  lambda10 >= L10: 0.00486

Report for threshold (L_{10}=0.0973 rightarrow E[v]=)$-144.02

b a
B (EP-negative) 0.99508
A (EP-positive) 0.00386

Simply as within the single-test case, by altering the check threshold within the case of utilizing each exams ( T_1+T_2 ), we’ve traded a better confidence in utilizing the check outcomes for sufferers who’re EP-positive (Ab = false damaging price decreases from about 1.75% → 0.39%) for a decrease confidence within the outcomes for sufferers who’re EP-negative (Ba = false optimistic price will increase from about 0.16% → 0.49%). This and the elevated value evaluation for false negatives signifies that the anticipated value will increase from $119.11 to $144.02 — which remains to be a lot better than the anticipated worth from the one-test value of $813.91.

In numeric phrases, for each 10 million sufferers, with 400 anticipated EP-positive sufferers, we are able to anticipate

  • 393 might be accurately identified as EP-positive, and seven might be misdiagnosed as EP-negative, with ( L_{10} = 1.1013 )
  • 398 might be accurately identified as EP-positive, and a pair of might be misdiagnosed as EP-negative, with ( L_{10} = 0.0973 )

(The cynical reader could conclude that, because the addition of check ( T_2 ) ends in a $670 lower in anticipated value over all potential sufferers, then the hospital must be charging $770 for check ( T_2 ), not $100.)

It’s price noting once more that we are able to by no means have good exams; there’s all the time some probability of the check being incorrect. The one solution to remove all false negatives is to extend the possibilities of false positives to 1. Alternative of thresholds is all the time a compromise.

One other factor to recollect is that actual conditions are seldom characterised completely by regular (Gaussian) distributions; the likelihood of occasions means out within the tails is often larger than Gaussian due to black-swan occasions.

Bear in mind: Por qué no los tres?

We’re nearly executed. We’ve simply proven that by having everybody take each exams, ( T_1 ) and ( T_2 ), we are able to maximize anticipated worth (reduce anticipated value) over all sufferers.

However that wasn’t the unique thought. Initially we have been going to do that:

  • Have everybody take the $1 check ( T_1 ), which ends up in a measurement ( x )
  • If ( x ge x_{H} ), diagnose as EP-positive, without having for check ( T_2 )
  • If ( x le x_{L} ), diagnose as EP-negative, without having for check ( T_2 )
  • If ( x_{L} < x < x_{H} ), we may have the affected person take the $100 check ( T_2 ), which ends up in a measurement ( y )

    • If ( lambda_{10}(x,y) ge L_{10} ), diagnose as EP-positive
    • If ( lambda_{10}(x,y) < L_{10} ), diagnose as EP-negative

the place ( lambda_{10}(x,y) = lambda(x,y) / ln 10 = log_{10} frac{p_A(x,y)}{p_B(x,y)}. )

Now we simply have to calculate thresholds ( x_H ) and ( x_L ). These are going to want to have very low false optimistic and false damaging charges, they usually’re there to catch the “apparent” circumstances with out the necessity to cost for (and look ahead to) check ( T_2 ).

We’ve got the identical type of calculation as earlier than. Let’s take into account ( x=x_H ). It must be chosen in order that with the proper threshold, there’s no change in anticipated worth if we alter the edge by a small quantity. (( partial E[v] / partial x_H = 0 )). We will do that by discovering ( x_H ) such that, inside a slender vary ( x_H < x < x_H+Delta x ), the anticipated worth is equal for each alternate options, particularly whether or not now we have them take check ( T_2 ), or diagnose them as EP-positive with out taking check ( T_2 ).

Primarily we’re figuring out thresholds ( x_H ) and ( x_L ) that, at every threshold, make the extra worth of knowledge gained from check ( T_2 ) (as measured by a change in anticipated worth) equal to the price of check ( T_2 ).

Do not forget that the entire likelihood of ( x_H < x < x_H+Delta x ) is ( (P_A p_A(x_H) + (1-P_A)p_B(x_H))Delta x ), damaged down into

  • ( P_A p_A(x_H) Delta x ) for EP-positive sufferers (A)
  • ( (1-P_A) p_B(x_H) Delta x ) for EP-negative sufferers (B)

Anticipated worth ( V_1 ) for diagnosing as EP-positive (a) with out taking check ( T_2 ), divided by ( Delta x ) so we don’t need to hold writing it:

$$V_1(x_H) = P_A v_{Aa}p_A(x_H) + (1-P_A) v_{Ba} p_B(x_H)$$

the place ( p_A(x), p_B(x) ) are Gaussian PDFs of the outcomes of check ( T_1 ).

Anticipated worth ( V_2 ) taking check ( T_2 ), which has worth ( v(T_2)=-$100 ):

V_2(x_0) &= v(T_2) + P_A E[v | x_0, A]p_A(x_0) + P_B E[v | x_0, B]p_B(x_0) cr
&= v(T_2) + P_A left(v_{Aa}P_{2a}(x_0 | A) + v_{Ab}P_{2b}(x_0 | A)proper) p_A(x_0) cr
&+ (1-P_A)left(v_{Ba} P_{2a}(x_0 | B) + v_{Bb}P_{2b}(x_0 | B)proper)p_B(x_0)

the place ( P_{2A}(x_0), P_{2B}(x_0) ) are the conditional chances of declaring the affected person as EP-positive or EP-negative primarily based on exams ( T_1, T_2 ), on condition that ( x=x_0 ). This occurs to be the numbers we used for numerical integration within the earlier part (the place we have been making all sufferers take exams ( T_1,T_2 )).

When now we have an optimum selection of threshold ( x_H ), the anticipated values might be equal: ( V_1(x_H) = V_2(x_H) ), as a result of there is no such thing as a benefit. If ( V_1(x_H) < V_2(x_H) ), then we haven’t chosen a very good threshold, and ( x_H ) must be decrease; if ( V_1(x_H) > V_2(x_H) ) then ( x_H ) must be larger. (Instance: suppose that ( x_H = 55, V_1(x_H) = -400, ) and ( V_2(x_H) = -250. ) The way in which we’ve outlined ( x_H ) is that any results of check ( T_1 ) the place ( x > x_H ), the worth of ( x ) is excessive sufficient that we’re higher off — in different phrases, the anticipated worth is meant to be larger — simply declaring prognosis ( a ) as an alternative of spending the additional $100 to get a consequence from check ( T_2 ), given its anticipated worth. In mathematical phrases, ( V_1(x) > V_2(x) ) so long as ( x > x_H ). However we are able to select ( x = x_H + delta ) with arbitrarily small ( delta ), and on this case now we have ( V_1(x_H + delta) > V_2(x_H+delta) ) which contradicts ( V_1(x_H) < V_2(x_H) ). Nitpicky mathematicians: which means that the anticipated values ( V_1(x_H) ) and ( V_2(x_H) ) as features of threshold ( x_H ) are steady there. The proof must be a trivial train for the reader, proper?)

So we simply want to seek out the fitting worth of ( x_H ) such that ( V_1(x_H) = V_2(x_H) ).

Presumably there’s a solution to decide a closed-form resolution right here, however I gained’t even hassle; we’ll simply consider it numerically.

We’ll additionally cowl the case the place we have to discover ( x_L ), the place we determine simply to make a prognosis ( b ) if ( x < x_L ) primarily based on ( T_1 ) alone, and the ensuing anticipated worth is

$$V_1(x_L) = P_A v_{Ab}p_A(x_L) + (1-P_A) v_{Bb} p_B(x_L),$$

in any other case if ( x ge x_L ), pay the $100 to take check ( T_2 ) with anticipated worth ( V_2(x_L) ).

Then we simply want to seek out ( x_L ) such that ( V_1(x_L) = V_2(x_L). )

Let’s get an thought of what these features appear to be for our instance.

def compute_value_densities(which_one, distA, distB, Pa_total, value_matrix, vT2, L):
    Returns a operate to compute worth densities V0, V1
    fxA = distA.slicefunc('x')
    fxB = distB.slicefunc('x')
    vAa = value_matrix[1,1]
    vBa = value_matrix[0,1]
    vAb = value_matrix[1,0]
    vBb = value_matrix[0,0]
    if which_one == 'H':
        vA1 = vAa
        vB1 = vBa
    elif which_one == 'L':
        vA1 = vAb
        vB1 = vBb
        elevate ValueError("which_one have to be 'H' or 'L'")
    normcdf = scipy.stats.norm.cdf
    C = distA.logpdf_coefficients - distB.logpdf_coefficients
    Q = Quadratic2D(*C)
    Qliss = Q.lissajous(L)
    xqr = Qliss.Rx
    xqmu = Qliss.x0
    # Discover the middle and radius of the contour lambda(x,y)=L
    def v1v2(x_0, verbose=False):
        wA, muA, sigmaA = fxA(x_0)
        wB, muB, sigmaB = fxB(x_0)
        # wA = likelihood density at x = x_0 given case A
        # wB = likelihood density at x = x_0 given case B
        # muA, sigmaA describe the pdf p(y | A,x=x0)
        # muB, sigmaB describe the pdf p(y | B,x=x0)
        if x_0 < xqmu - xqr or x_0 > xqmu + xqr:
            # x is excessive sufficient that we all the time diagnose as "b"
            P2Aa = 0
            P2Ab = 1
            P2Ba = 0
            P2Bb = 1
            # Right here we have to discover the y-value thresholds
            theta = np.arccos((x_0-xqmu)/xqr)
            x1,y1 = Qliss(theta)
            x2,y2 = Qliss(-theta)
            assert np.abs(x_0-x1) < 1e-10*xqr, (x_0,x1,x2)
            assert np.abs(x_0-x2) < 1e-10*xqr, (x_0,x1,x2)
            if y1 > y2:
                y1,y2 = y2,y1
            assert np.abs(Q(x_0,y1) - L) < 1e-10
            assert np.abs(Q(x_0,y2) - L) < 1e-10
            # Diagnose as "a" if between the thresholds, in any other case "b"
            P2Aa = normcdf(y2, muA, sigmaA) - normcdf(y1, muA, sigmaA)
            P2Ab = 1-P2Aa
            P2Ba = normcdf(y2, muB, sigmaB) - normcdf(y1, muB, sigmaB)
            P2Bb = 1-P2Ba
        # anticipated worth given the affected person has outcomes of each exams
        # over the complete vary of y
        vA2 = vAa*P2Aa+vAb*P2Ab    # given A, x_0
        vB2 = vBa*P2Ba+vBb*P2Bb    # given B, x_0
        # Bayes' rule for conditional likelihood of A and B given x_0
        PA = (Pa_total*wA)/(Pa_total*wA + (1-Pa_total)*wB)
        PB = 1-PA
        v1 =       PA*vA1 + PB*vB1
        v2 = vT2 + PA*vA2 + PB*vB2
        if verbose:
            print which_one, x_0
            print "PA|x0=",PA
            print vAa,vAb,vBa,vBb
            print P2Aa, P2Ab, P2Ba, P2Bb
            print "T1 solely", vA1,vB1
            print "T1+T2  ", vA2,vB2
            print "v1=%g v2=%g v2-v1=%g" % (v1,v2,v2-v1)
        return v1,v2
    return v1v2

Pa_total = 40e-6
value_matrix_T1 = np.array([[0,-5000],[-1e7, -1e5]])
vT2 = -100
L = compute_optimal_L(value_matrix_T1 + vT2, Pa_total)
distA2 = Gaussian2D(mu_x=63,mu_y=57,sigma_x=5.3,sigma_y=4.1,rho=0.91,title="A (EP-positive)",id='A',coloration="purple")
distB2 = Gaussian2D(mu_x=48,mu_y=36,sigma_x=5.9,sigma_y=5.2,rho=0.84,title="B (EP-negative)",id='B',coloration="#8888ff")

print "For L10=%.4f:" % (L/np.log(10)) 
for which in 'HL':
    print ""
    v1v2 = compute_value_densities(which, distA, distB, Pa_total, value_matrix_T1, vT2, L)
    for x_0 in np.arange(25,100.1,5):
        v1,v2 = v1v2(x_0)
        print "x_percents=%.1f, V1=%.4g, V2=%.4g, V2-V1=%.4g" % (which, x_0,v1,v2,v2-v1)
For L10=1.1013:

x_H=25.0, V1=-5000, V2=-100, V2-V1=4900
x_H=30.0, V1=-5000, V2=-100, V2-V1=4900
x_H=35.0, V1=-5000, V2=-100.5, V2-V1=4900
x_H=40.0, V1=-5000, V2=-104.2, V2-V1=4896
x_H=45.0, V1=-5001, V2=-122.2, V2-V1=4879
x_H=50.0, V1=-5011, V2=-183.7, V2-V1=4828
x_H=55.0, V1=-5107, V2=-394.5, V2-V1=4713
x_H=60.0, V1=-5840, V2=-1354, V2-V1=4487
x_H=65.0, V1=-1.033e+04, V2=-6326, V2-V1=4008
x_H=70.0, V1=-2.878e+04, V2=-2.588e+04, V2-V1=2902
x_H=75.0, V1=-6.315e+04, V2=-6.185e+04, V2-V1=1301
x_H=80.0, V1=-8.695e+04, V2=-8.661e+04, V2-V1=339.4
x_H=85.0, V1=-9.569e+04, V2=-9.567e+04, V2-V1=26.9
x_H=90.0, V1=-9.843e+04, V2=-9.849e+04, V2-V1=-59.82
x_H=95.0, V1=-9.933e+04, V2=-9.942e+04, V2-V1=-85.24
x_H=100.0, V1=-9.967e+04, V2=-9.976e+04, V2-V1=-93.59

x_L=25.0, V1=-0.001244, V2=-100, V2-V1=-100
x_L=30.0, V1=-0.0276, V2=-100, V2-V1=-100
x_L=35.0, V1=-0.5158, V2=-100.5, V2-V1=-99.95
x_L=40.0, V1=-8.115, V2=-104.2, V2-V1=-96.09
x_L=45.0, V1=-107.5, V2=-122.2, V2-V1=-14.63
x_L=50.0, V1=-1200, V2=-183.7, V2-V1=1016
x_L=55.0, V1=-1.126e+04, V2=-394.5, V2-V1=1.087e+04
x_L=60.0, V1=-8.846e+04, V2=-1354, V2-V1=8.711e+04
x_L=65.0, V1=-5.615e+05, V2=-6326, V2-V1=5.551e+05
x_L=70.0, V1=-2.503e+06, V2=-2.588e+04, V2-V1=2.477e+06
x_L=75.0, V1=-6.121e+06, V2=-6.185e+04, V2-V1=6.059e+06
x_L=80.0, V1=-8.627e+06, V2=-8.661e+04, V2-V1=8.54e+06
x_L=85.0, V1=-9.547e+06, V2=-9.567e+04, V2-V1=9.451e+06
x_L=90.0, V1=-9.835e+06, V2=-9.849e+04, V2-V1=9.736e+06
x_L=95.0, V1=-9.93e+06, V2=-9.942e+04, V2-V1=9.83e+06
x_L=100.0, V1=-9.965e+06, V2=-9.976e+04, V2-V1=9.865e+06
import matplotlib.gridspec

def fVdistort(V):
    return -np.log(-np.array(V+ofs))

yticks0 =np.array([1,2,5])
yticks = -np.hstack([yticks0 * 10**k for k in xrange(-1,7)])

for which in 'HL':
    fig = plt.determine(figsize=(6,6))
    gs = matplotlib.gridspec.GridSpec(2,1,height_ratios=[2,1],hspace=0.1)
    x = np.arange(20,100,0.2)
    fv1v2 = compute_value_densities(which, distA, distB, Pa_total, value_matrix_T1, vT2, L)
    v1v2 = np.array([fv1v2(xi) for xi in x])
    vlim = np.array([max(-1e6,v1v2.min()*1.01),
    f = fVdistort
    ax.plot(x,f(v1v2[:,0]),label="T1 solely (declare %s if $xpercents$)" %
            (('"a"', '>x_H') if which == 'H' else ('"b"', '<x_L')))
    ax.set_yticklabels('%.0f' % y for y in yticks)
    ax.legend(labelspacing=0, fontsize=10,
              loc="decrease left" if which=='H' else 'higher proper')
    [label.set_visible(False) for label in ax.get_xticklabels()]
    ax2=fig.add_subplot(gs[1], sharex=ax)
    ax2.set_ylabel('$Delta E[v]$')
    ax2.set_xlabel('$x_percents$' % which, fontsize=12)

It seems like for this case (( L_{10}=1.1013 )), we must always select ( x_L approx 45 ) and ( x_H approx 86 ).

These edge circumstances the place ( x < x_L ) or ( x > x_H ) don’t save some huge cash, at finest simply the $100 ( T_2 ) check value… so a not-quite-as-optimal however easier case can be to all the time run each exams. Nonetheless, there’s a giant distinction between going to the physician and paying $1 slightly than $101… whereas when you’ve paid a $100,000 hospital invoice, what’s an additional $100 amongst associates?

Between these thresholds, the place check ( T_1 ) is type of unhelpful, the good thing about operating each exams is gigantic: for EP-positive sufferers we can assist reduce these pesky false negatives, saving hospitals hundreds of thousands in malpractice costs and serving to those that would in any other case have grieving households; for EP-negative sufferers we are able to restrict the false positives and save them the $5000 and anguish of a annoying hospital keep.

Placing all of it collectively

Now we are able to present our full check protocol on one graph and one chart. Beneath the coloured highlights present 4 areas:

  • blue: ( b_1 ) — EP-negative prognosis after taking check ( T_1 ) solely, with ( x<x_L )
  • inexperienced: ( b_2 ) — EP-negative prognosis after taking exams ( T_1, T_2 ), with ( x_L le x le x_H ) and ( lambda_{10} < L_{10} )
  • yellow: ( a_2 ) — EP-positive prognosis after taking exams ( T_1, T_2 ), with ( x_L le x le x_H ) and ( lambda_{10} ge L_{10} )
  • purple: ( a_1 ) — EP-positive prognosis after taking exams ( T_1 ) solely, with ( x > x_H )
import matplotlib.patches as patches
import matplotlib.path
import scipy.optimize
Path = matplotlib.path.Path

def show_complete_chart(xydistA, xydistB, Q, L, Pa_total, value_matrix_T1, vT2,
    fig = plt.determine()
    ax = fig.add_subplot(1,1,1)
    xlim = (0,100)
    ylim = (0,100)
    separation_plot(xydistA, xydistB, Q, L, ax=ax, xlim=xlim, ylim=ylim)
    # Remedy for xL and xH
    _,_,distA = xydistA
    _,_,distB = xydistB
    v1v2 = [compute_value_densities(which, distA, distB, Pa_total,
                                    value_matrix_T1, vT2, L)
            for which in 'LH']
    def fdelta(f):
        def g(x):
            y1,y2 = f(x)
            return y1-y2
        return g    
    xL = scipy.optimize.brentq(fdelta(v1v2[0]), 0, 100)
    xH = scipy.optimize.brentq(fdelta(v1v2[1]), 0, 100)
    # Present the complete matrix of potentialities:
    # compute 2x4 confusion matrix
    C = []
    for dist in [distB, distA]:
        distx = dist.venture('x')
        pa2,pb2 = estimate_separation_numerical(dist, Q, L, xL, xH, 
        row = [distx.cdf(xL), pb2, pa2, 1-distx.cdf(xH)]
    # compute 2x4 worth matrix
    V = value_matrix_T1.repeat(2,axis=1)
    V[:,1:3] += vT2

                       threshold='x_L=%.3f, x_H=%.3f, L_{10}=%.3f'

    # Spotlight every of the areas
    x0,x1 = xlim
    y0,y1 = ylim
    # space b1: x < xL
    rect = patches.Rectangle((x0,y0),xL-x0,y1-y0,linewidth=0,edgecolor=None,facecolor="#8888ff",alpha=0.1)
    # space a1: x > xH
    rect = patches.Rectangle((xH,y0),x1-xH,y1-y0,linewidth=0,edgecolor=None,facecolor="purple",alpha=0.1)
    for x in [xL,xH]:
    # space a2: lambda(x,y) > L 
    xc,yc = Q.contour(L)
    ii = (xc > xL-10) & (xc < xH + 10)
    xc = xc[ii]
    yc = yc[ii]
    xc = np.most(xc,xL)
    xc = np.minimal(xc,xH)
    poly2a = patches.Polygon(np.vstack([xc,yc]).T, edgecolor=None, facecolor="yellow",alpha=0.5)
    # space b2: lambda(x,y) < L     
    xy = []
    op = []
    i = 0
    for x,y in zip(xc,yc):
        i += 1
        op.append(Path.MOVETO if i == 1 else Path.LINETO)
    xy += [(xL,y0),(xL,y1),(xH,y1),(xH,y0),(0,0)]
    op += [Path.MOVETO, Path.LINETO, Path.LINETO, Path.LINETO, Path.CLOSEPOLY]
    patch2b = patches.PathPatch(Path(xy,op), edgecolor=None, facecolor="#00ff00",alpha=0.1)

    # add labels
    type = {'fontsize': 28, 'ha':'heart'}
    ax.textual content((x0+xL)/2,y0*0.3+y1*0.7,'$b_1$', **type)
    ax.textual content(xc.imply(), yc.imply(), '$a_2$', **type)
    a = 0.3 if yc.imply() > (x0+x1)/2 else 0.7
    yb2 = a*y1 + (1-a)*y0
    ax.textual content(xc.imply(), yb2, '$b_2$',**type)
    xa1 = (xH + x1) / 2 
    ya1 = max(30, min(90, Q.constrain(x=xa1).x0))
    ax.textual content(xa1,ya1,'$a_1$',**type)
    if return_info:
        C = np.array(C)
        J = C * [[1-Pa_total],[Pa_total]]
        return dict(C=C,J=J,V=V,xL=xL,xH=xH,L=L)

value_matrix_T1 = np.array([[0,-5000],[-1e7, -1e5]])
value_matrix_T2 = value_matrix_T1 - 100
L = compute_optimal_L(value_matrix_T2, Pa_total)
info37 = show_complete_chart((xA,yA,distA),(xB,yB,distB), Q, L, Pa_total, 
                            value_matrix_T1, vT2, return_info=True)

Report for threshold (x_L=45.287, x_H=85.992, L_{10}=1.101 rightarrow E[v]=)$-46.88

b1 b2 a2 a1
B (EP-negative) 0.81491
A (EP-positive) 0.03343

We will present the identical data however with the false damaging value (Ab = wrongly identified EP-negative) at $100 million:

value_matrix_T1 = np.array([[0,-5000],[-1e8, -1e5]])
value_matrix_T2 = value_matrix_T1 - 100
L = compute_optimal_L(value_matrix_T2, Pa_total)
info38 = show_complete_chart((xA,yA,distA),(xB,yB,distB), Q, L, Pa_total, 
                            value_matrix_T1, vT2, return_info=True)

Report for threshold (x_L=40.749, x_H=85.108, L_{10}=0.097 rightarrow E[v]=)$-99.60

b1 b2 a2 a1
B (EP-negative) 0.55051
A (EP-positive) 0.00358

Do we want check ( T_1 )?

If including check ( T_2 ) is so a lot better than check ( T_1 ) alone, why do we want check ( T_1 ) in any respect? What if we simply gave everybody check ( T_2 )?

y = np.arange(0,100,0.1)
distA_T2 = distA.venture('y')
distB_T2 = distB.venture('y')
show_binary_pdf(distA_T2, distB_T2, y, xlabel="$T_2$ check consequence $y$")

for false_negative_cost in [10e6, 100e6]:
    value_matrix_T2 = np.array([[0,-5000],[-false_negative_cost, -1e5]]) - 100
    threshold_info = find_threshold(distB_T2, distA_T2, Pa_total, value_matrix_T2)
    y0 = [yval for yval,_ in threshold_info if yval > 20 and yval < 80][0]

    C = analyze_binary(distB_T2, distA_T2, y0)
    print "False damaging value = $%.0fM" % (false_negative_cost/1e6)
    show(show_binary_matrix(C, 'y_0=%.2f' % y0, [distB_T2, distA_T2], 'ba', Pa_total, value_matrix_T2,
    information = gather_info(C,value_matrix_T2,Pa_total,y0=y0)
    if false_negative_cost == 10e6:
        info47 = information
        info48 = information
False damaging value = $10M

Report for threshold (y_0=50.14 rightarrow E[v]=)$-139.03

b a
B (EP-negative) 0.99673
A (EP-positive) 0.04717
False damaging value = $100M

Report for threshold (y_0=47.73 rightarrow E[v]=)$-211.68

b a
B (EP-negative) 0.98795
A (EP-positive) 0.01187

Hmm. That appears higher than the ( T_1 ) check… however the confusion matrix doesn’t appear pretty much as good as utilizing each exams.

d prime ( (d’) ): Are two exams are all the time higher than one?

Why would figuring out a prognosis primarily based on each exams ( T_1 ) and ( T_2 ) be higher than from both check alone?

Let’s graph the PDFs of three completely different measurements:

  • ( x ) (the results of check ( T_1 ))
  • ( y ) (the results of check ( T_2 ))
  • ( u = -0.88x + 1.88y ) (a linear mixture of the 2 measurements)

We’ll additionally calculate a metric for every of the three measurements,

$$d’ = frac{mu_A – mu_B}{sqrt{frac{1}{2}(sigma_A{}^2 + sigma_B{}^2)}}$$

which is a measure of the “distinguishability” between two populations which every have regular distributions. Primarily it’s a unitless “separation coefficient” that measures the distances between the means as a a number of of the “common” commonplace deviation. It is usually invariant below scaling and translation — if we work out the worth of ( d’ ) for some measurement ( x ), then any derived measurement ( u = ax + c ) has the identical worth for ( d’ ) so long as ( a ne 0 ).

(We haven’t talked about derived measurements like ( u ), however for a 2-D Gaussian distribution, if ( u=ax+by+c ) then:

mu_u &= E[u] = aE[x]+bE[y] = amu_x + bmu_y + ccr
sigma_u{}^2 &= E[(u-mu_u)^2] cr
&= E[a^2(x-mu_x)^2 + 2ab(x-mu_x)(y-mu_y) + b^2(y-mu_y)^2] cr
&= a^2E[(x-mu_x)^2] + 2abE[(x-mu_x)(y-mu_y)] + b^2E[(y-mu_y)^2] cr
&= a^2 sigma_x{}^2 + 2abrhosigma_xsigma_y + b^2sigma_y{}^2

or, alternatively utilizing matrix notation, ( sigma_u{}^2 = mathrm{v}^T mathrm{S} mathrm{v} ) the place ( mathrm{v} = start{bmatrix}a& bend{bmatrix}^T ) is the vector of weighting coefficients, and ( mathrm{S} = operatorname{cov}(x,y) = start{bmatrix}sigma_x{}^2 & rhosigma_xsigma_y cr rhosigma_xsigma_y & sigma_y{}^2end{bmatrix} ). Yep, there’s the covariance matrix once more.)

def calc_dprime(distA,distB,projection):
    """ calculates dprime for 2 distributions,
    given a projection P = [cx,cy]
    that defines u = cx*x + cy*y
    distAp = distA.venture(projection)
    distBp = distB.venture(projection)
    return ( -
              (distAp.sigma**2 + distBp.sigma**2)/2.0

def show_dprime(distA, distB, a,b):
    print "u=%.6fx%+.6fy" % (a,b)
    x = np.arange(0,100,0.1)
    fig = plt.determine()
    ax = fig.add_subplot(1,1,1)
    for projname, projection, linestyle in [
        distAp = distA.venture(projection)
        distBp = distB.venture(projection)
        dprime = calc_dprime(distA, distB, projection)
        print "dprime(%s)=%.4f" % (projname,dprime)
        for dist in [distAp,distBp]:
                    label="$%s$: %s, $mu=$%.1f, $sigma=$%.2f" % 
                        (projname,,, dist.sigma))
    ax.set_ylabel('likelihood density')
    ax.set_xlabel('measurement $(x,y,u)$')
    ax.legend(labelspacing=0, fontsize=10,loc="higher left");
show_dprime(distA, distB, -0.88,1.88)

We will get a greater separation between alternate options, as measured by ( d’ ), via this linear mixture of ( x ) and ( y ) than by both one alone. What’s occurring, and the place did the equation ( u=-0.88x + 1.88y ) come from?

Can we do higher than that? How about ( u=-0.98x + 1.98y )? or ( u=-0.78x + 1.78y )?

show_dprime(distA, distB, -0.98,1.98)
show_dprime(distA, distB, -0.78,1.78)

Hmm. These aren’t fairly pretty much as good; the worth for ( d’ ) is decrease. How do we all know how you can maximize ( d’ )?

Start grungy algebra

There doesn’t appear to be a easy intuitive solution to discover the most effective projection. At first I considered modeling this as ( v = x costheta + y sintheta ) for some angle ( theta ). However this didn’t produce any simple reply. I muddled my solution to a solution by on the lookout for patterns that helped to cancel out a few of the grunginess.

One solution to maximize ( d’ ) is to put in writing it as a operate of ( theta ) with ( v=ax+by, a = a_1cos theta+a_2sintheta, b = b_1costheta+b_2sintheta ) for some handy ( a_1, a_2, b_1, b_2 ) to make the mathematics good. We’re going to determine ( d’ ) as a operate of ( a,b ) generally first:

d’ &= frac{mu_{vA} – mu_{vB}}{sqrt{frac{1}{2}(sigma_{vA}{}^2 + sigma_{vB}{}^2)}} cr
&= frac{a(mu_{xA}-mu_{xB})+b(mu_{yA} – mu_{yB})}{sqrt{frac{1}{2}(a^2(sigma_{xA}^2 + sigma_{xB}^2) + 2ab(rho_Asigma_{xA}sigma_{yA} + rho_Bsigma_{xB}sigma_{yB}) + b^2(sigma_{yA}^2+sigma_{yB}^2))}} cr
&= frac{a(mu_{xA}-mu_{xB})+b(mu_{yA} – mu_{yB})}{sqrt{frac{1}{2}f(a,b)}} cr

with ( f(a,b) = a^2(sigma_{xA}^2 + sigma_{xB}^2) + 2ab(rho_Asigma_{xA}sigma_{yA} + rho_Bsigma_{xB}sigma_{yB}) + b^2(sigma_{yA}^2+sigma_{yB}^2). )


Anyway, if we are able to make the denominator fixed as a operate of ( theta ), then the numerator is straightforward to maximise.

If we outline

K_x &= sqrt{sigma_{xA}^2 + sigma_{xB}^2} cr
K_y &= sqrt{sigma_{yA}^2 + sigma_{yB}^2} cr
R &= frac{rho_Asigma_{xA}sigma_{yA} + rho_Bsigma_{xB}sigma_{yB}}{K_xK_y}

then ( f(a,b) = a^2K_x{}^2 + 2abRK_xK_y + b^2K_y{}^2 ) which is simpler to put in writing than having to hold round all these ( mu ) and ( sigma ) phrases.

If we let ( a = frac{1}{sqrt{2}K_x}(c cos theta + s sintheta) ) and ( b = frac{1}{sqrt{2}K_y}(c cos theta – ssintheta) ) then

f(a,b) &= frac{1}{2}(c^2 cos^2theta + s^2sin^2theta + 2cscosthetasintheta) cr
&+ frac{1}{2}(c^2 cos^2theta + s^2sin^2theta – 2cscosthetasintheta) cr
&+R(c^2 cos^2theta – s^2sin^2theta) cr
&= (1+R)c^2 cos^2theta + (1-R)s^2sin^2theta

which is unbiased of ( theta ) if ( (1+R)c^2 = (1-R)s^2 ). If we let ( c = cos phi ) and ( s=sin phi ) then some good issues all fall out:

  • ( f(a,b) ) is fixed if ( tan^2phi = frac{1+R}{1-R} ), in different phrases ( phi = tan^{-1} sqrt{frac{1+R}{1-R}} )
  • ( c = cosphi = sqrt{frac{1-R}{2}} ) (trace: use the identities ( sec^2 theta = tan^2 theta + 1 ) and ( cos theta = 1/sec theta ))
  • ( s = sinphi = sqrt{frac{1+R}{2}} )
  • ( f(a,b) = (1-R^2)/2 )
  • ( a = frac{1}{sqrt{2}K_x}cos (theta – phi) = frac{1}{sqrt{2}K_x}(cos phi cos theta + sinphi sintheta)= frac{1}{2K_x}(sqrt{1-R} cos theta + sqrt{1+R} sintheta) )
  • ( b = frac{1}{sqrt{2}K_y}cos (theta + phi) = frac{1}{sqrt{2}K_y}(cos phi cos theta – sinphi sintheta)= frac{1}{2K_y}(sqrt{1-R} cos theta – sqrt{1+R} sintheta) )

and when all is alleged and executed we get

d’ &= frac{a(mu_{xA}-mu_{xB})+b(mu_{yA} – mu_{yB})}{sqrt{frac{1}{2}f(a,b)}} cr
&= frac{a(mu_{xA}-mu_{xB})+b(mu_{yA} – mu_{yB})}{frac{1}{2}sqrt{1-R^2}} cr
&= K_c costheta + K_s sintheta


delta_x &= frac{mu_{xA}-mu_{xB}}{K_x} = frac{mu_{xA}-mu_{xB}}{sqrt{sigma_{xA}^2 + sigma_{xB}^2}} cr
delta_y &= frac{mu_{yA} – mu_{yB}}{K_y} = frac{mu_{yA} – mu_{yB}}{sqrt{sigma_{yA}^2 + sigma_{yB}^2}} cr
K_c &= sqrt{frac{1-R}{1-R^2}} left(delta_x + delta_y proper)
&= frac{1}{sqrt{1+R}} left(delta_x + delta_y proper) cr
K_s &= sqrt{frac{1+R}{1-R^2}} left(delta_x – delta_y proper)
&= frac{1}{sqrt{1-R}} left(delta_x – delta_y proper) cr

Then ( d’ ) has a most worth of ( D = sqrt{K_c{}^2 + K_s{}^2} ) at ( theta = tan^{-1}dfrac{K_s}{K_c} ), the place ( cos theta = dfrac{K_c}{sqrt{K_s^2 + K_c^2}} ) and ( sin theta = dfrac{K_s}{sqrt{K_s^2 + K_c^2}}. )

Plugging into ( a ) and ( b ) we get

a &= frac{1}{2K_x}(sqrt{1-R} cos theta + sqrt{1+R} sintheta)cr
&= frac{1}{2K_x}left(frac{(1-R)(delta_x+delta_y)}{sqrt{1-R^2}}+frac{(1+R)(delta_x-delta_y)}{sqrt{1-R^2}}proper)cdotfrac{1}{sqrt{K_s^2 + K_c^2}} cr
&= frac{delta_x – Rdelta_y}{K_xDsqrt{1-R^2}}cr
b &= frac{1}{2K_y}(sqrt{1-R} cos theta – sqrt{1+R} sintheta)cr
&= frac{1}{2K_y}left(frac{(1-R)(delta_x+delta_y)}{sqrt{1-R^2}}-frac{(1+R)(delta_x-delta_y)}{sqrt{1-R^2}}proper)cdotfrac{1}{sqrt{K_s^2 + K_c^2}} cr
&= frac{-Rdelta_x + delta_y}{K_yDsqrt{1-R^2}}

We will additionally clear up ( D ) by way of ( delta_x, delta_y, ) and ( R ) to get

D &= frac{1}{sqrt{1-R^2}}sqrt{(1-R)(delta_x+delta_y)^2 + (1+R)(delta_x-delta_y)^2} cr
&= sqrt{frac{2delta_x^2 -4Rdelta_xdelta_y + 2delta_y^2}{1-R^2}} cr
a &= frac{delta_x – Rdelta_y}{K_xsqrt{2delta_x^2 -4Rdelta_xdelta_y + 2delta_y^2}} cr
b &= frac{-Rdelta_x + delta_y}{K_ysqrt{2delta_x^2 -4Rdelta_xdelta_y + 2delta_y^2}}

Let’s attempt it!

Kx = np.hypot(distA.sigma_x,distB.sigma_x)
Ky = np.hypot(distA.sigma_y,distB.sigma_y)
R = (distA.rho*distA.sigma_x*distA.sigma_y + distB.rho*distB.sigma_x*distB.sigma_y)/Kx/Ky
Kx, Ky, R
(7.9309520235593407, 6.6219332524573211, 0.86723211589363869)
dmux = (distA.mu_x - distB.mu_x)/Kx
dmuy = (distA.mu_y - distB.mu_y)/Ky
Kc = (dmux + dmuy)/np.sqrt(1+R)
Ks = (dmux - dmuy)/np.sqrt(1-R)
(3.7048851247525367, -3.5127584699841927)
theta = np.arctan(Ks/Kc)
a = 1.0/2/Kx*(np.sqrt(1-R)*np.cos(theta) + np.sqrt(1+R)*np.sin(theta) )
b = 1.0/2/Ky*(np.sqrt(1-R)*np.cos(theta) - np.sqrt(1+R)*np.sin(theta) )
dprime = np.hypot(Kc,Ks)
dprime2 = Kc*np.cos(theta) + Ks*np.sin(theta)
assert abs(dprime - dprime2) < 1e-7
delta_x = (distA.mu_x - distB.mu_x)/np.hypot(distA.sigma_x,distB.sigma_x)
delta_y = (distA.mu_y - distB.mu_y)/np.hypot(distA.sigma_y,distB.sigma_y)
dd = np.sqrt(2*delta_x**2 - 4*R*delta_x*delta_y + 2*delta_y**2)
dprime3 = dd/np.sqrt(1-R*R)
assert abs(dprime - dprime3) < 1e-7
a2 = (delta_x - R*delta_y)/Kx/dd
b2 = (-R*delta_x + delta_y)/Ky/dd
assert abs(a-a2) < 1e-7
assert abs(b-b2) < 1e-7
print "theta=%.6f a=%.6f b=%.6f d'=%.6f" % (theta, a, b, dprime)
# scale a,b such that their sum is 1.0
print "   additionally a=%.6f b=%.6f is an answer with a+b=1" % (a/(a+b), b/(a+b))
theta=-0.758785 a=-0.042603 b=0.090955 d'=5.105453
   additionally a=-0.881106 b=1.881106 is an answer with a+b=1

And out pops ( uapprox -0.88x + 1.88y ).

Finish grungy algebra

What if we use ( u=-0.8811x + 1.8811y ) as a solution to mix the outcomes of exams ( T_1 ) and ( T_2 )? Here’s a superposition of strains with fixed ( u ):

# Strains of fixed u
fig = plt.determine()
ax = fig.add_subplot(1,1,1)
separation_plot((xA,yA,distA),(xB,yB,distB), Q, L, ax=ax)
xmax = ax.get_xlim()[1]
ymax = ax.get_ylim()[1]
ua = a/(a+b)
ub = b/(a+b)
for u in np.arange(ua*xmax,ub*ymax+0.01,10):
    x1 = min(xmax,(u-ub*ymax)/ua)
    x0 = max(0,u/ua)
    x = np.array([x0,x1])

To summarize:

R &= frac{rho_Asigma_{xA}sigma_{yA} + rho_Bsigma_{xB}sigma_{yB}}{sqrt{(sigma_{xA}^2 + sigma_{xB}^2)(sigma_{yA}^2 + sigma_{yB}^2)}} cr
delta_x &= frac{mu_{xA}-mu_{xB}}{sqrt{sigma_{xA}^2 + sigma_{xB}^2}} cr
delta_y &= frac{mu_{yA} – mu_{yB}}{sqrt{sigma_{yA}^2 + sigma_{yB}^2}} cr
max d’ = D &= sqrt{frac{2delta_x^2 -4Rdelta_xdelta_y + 2delta_y^2}{1-R^2}} cr
a&= frac{delta_x – Rdelta_y}{K_xsqrt{2delta_x^2 -4Rdelta_xdelta_y + 2delta_y^2}}cr
b&= frac{-Rdelta_x + delta_y}{K_ysqrt{2delta_x^2 -4Rdelta_xdelta_y + 2delta_y^2}}cr

and we are able to calculate a brand new derived measurement ( u=ax+by ) which has separation coefficient ( d’ ) between its distributions below the situations ( A ) and ( B ).

u = np.arange(0,100,0.1)
projAB = [-0.8811, 1.8811]
distA_T1T2lin = distA.venture(projAB)
distB_T1T2lin = distB.venture(projAB)
show_binary_pdf(distA_T1T2lin, distB_T1T2lin, y, 
                xlabel="$u = %.4fx + %.4fy$" % tuple(projAB))

for false_negative_cost in [10e6, 100e6]:
    value_matrix_T2 = np.array([[0,-5000],[-false_negative_cost, -1e5]]) - 100
    threshold_info = find_threshold(distB_T1T2lin, distA_T1T2lin, Pa_total, value_matrix_T2)
    u0 = [uval for uval,_ in threshold_info if uval > 20 and uval < 80][0]

    C = analyze_binary(distB_T1T2lin, distA_T1T2lin, u0)
    print "False damaging value = $%.0fM" % (false_negative_cost/1e6)
    show(show_binary_matrix(C, 'u_0=ax+by=%.2f' % u0,
                               [distB_T1T2lin, distA_T1T2lin],
                               'ba', Pa_total, value_matrix_T2,
    information = gather_info(C,value_matrix_T2,Pa_total,u0=u0)
    if false_negative_cost == 10e6:
        info57 = information
        info58 = information
False damaging value = $10M

Report for threshold (u_0=ax+by=50.42 rightarrow E[v]=)$-119.26

b a
B (EP-negative) 0.99835
A (EP-positive) 0.01771
False damaging value = $100M

Report for threshold (u_0=ax+by=48.22 rightarrow E[v]=)$-144.54

b a
B (EP-negative) 0.99504
A (EP-positive) 0.00394

One closing subtlety

Earlier than we wrap up the mathematics, there’s yet one more factor that’s price a quick point out.

Once we use each exams with the quadratic operate ( lambda(x,y) ), there’s a type of humorous area we haven’t talked about. Look on the graph beneath, at level ( P = (x=40, y=70) ):

fig = plt.determine()
ax = fig.add_subplot(1,1,1)
separation_plot((xA,yA,distA),(xB,yB,distB), Q, L, ax=ax)
Ptext = Coordinate2D(P.x-10,P.y)
ax.textual content(Ptext.x,Ptext.y,'$P$',fontsize=20,

This level is exterior the contour ( lambda_{10} > L_{10} ), in order that tells us we must always give a prognosis of EP-negative. However this level is nearer to the likelihood “cloud” of EP-positive. Why?

for dist in [distA, distB]:
    print("Chance density at P of %s = %.3g" 
          % (dist.title, dist.pdf(P.x,P.y)))
Chance density at P of A (EP-positive) = 6.28e-46
Gaussian(mu_x=55, mu_y=57, sigma_x=5.3, sigma_y=4.1, rho=0.91, title="A (EP-positive)", id='A', coloration="purple")
Chance density at P of B (EP-negative) = 2.8e-34
Gaussian(mu_x=40, mu_y=36, sigma_x=5.9, sigma_y=5.2, rho=0.84, title="B (EP-negative)", id='B', coloration="#8888ff")

The likelihood density is smaller at P for the EP-positive case, despite the fact that this level is nearer to the corresponding likelihood distribution. It’s because the distribution is narrower; ( sigma_x ) and ( sigma_y ) are each smaller for the EP-positive case.

As a thought experiment, think about the next distributions A and B, the place B could be very huge (( sigma=5 )) in comparison with A (( sigma = 0.5 )):

dist_wide = Gaussian1D(20,5,'B (huge)',id='B',coloration="inexperienced")
dist_narrow = Gaussian1D(25,0.5,'A (slender)',id='A',coloration="purple")
x = np.arange(0,40,0.1)
show_binary_pdf(dist_wide, dist_narrow, x)

On this case, if we take some pattern measurement ( x ), a classification of ( A ) is smart provided that the measured worth ( x ) is close to ( A )’s imply worth ( mu=25 ), say from 24 to 26. Exterior that vary, a classification of ( B ) is smart, not as a result of the measurement is nearer to the imply of ( B ), however as a result of the anticipated likelihood of ( B ) given ( x ) is bigger. This holds true even for values inside just a few commonplace deviations from the imply, like ( x=28 = mu_B + 1.6sigma_B ), as a result of the distribution of ( A ) is so slender.

We’ve been proposing exams the place there’s a single threshold — for instance diagnose as ( a ) if ( x > x_0 ), in any other case diagnose as ( b ) — however there are pretty easy circumstances the place two thresholds are required. Actually, that is true generally when the usual deviations are unequal; in the event you get far sufficient away from the means, the likelihood of the broader distribution is bigger.

You may suppose, properly, that’s type of unusual, if I’m utilizing the identical piece of kit to measure all of the pattern knowledge, why would the usual deviations be completely different? A digital multimeter measuring voltage, for instance, has some inherent uncertainty. However generally the measurement variation is because of variations within the populations themselves. For instance, take into account the wingspans of two populations of birds, the place ( A ) consists of birds which can be pigeons and ( B ) consists of birds that aren’t pigeons. The ( B ) inhabitants may have a wider vary of measurements just because this inhabitants is extra heterogeneous.

Please word, by the best way, that the Python features I wrote to research the bivariate Gaussian state of affairs make the belief that the usual deviations are unequal and the log-likelihood ratio ( lambda(x,y) ) is both concave upwards or concave downwards — in different phrases, the ( A ) distribution has decrease values of each ( sigma_x ) and ( sigma_y ) than the ( B ) distribution, or it has larger values of each ( sigma_x ) and ( sigma_y ). In these circumstances, the contours of fixed ( lambda ) are ellipses. Generally, they might be another conic part (strains, parabolas, hyperbolas) however I didn’t really feel like making an attempt to attain correctness in all circumstances, for the needs of this text.

So which check is finest?

Let’s summarize the exams we got here up with. We’ve got 10 completely different exams, particularly every of the next with false-negative prices of $10 million and $100 million:

  1. check ( T_1 ) solely
  2. exams ( T_1 ) and ( T_2 ) – quadratic operate of ( x,y )
  3. check ( T_1 ) first, then ( T_2 ) if wanted
  4. check ( T_2 ) solely
  5. exams ( T_1 ) and ( T_2 ) – linear operate of ( x,y )
import pandas as pd

def present_info(information):
    C = information['C']
    J = information['J']
    V = information['V']
    if 'x0' in information:
        threshold = '( x_0 = %.2f )' % information['x0']
        t = 1
        Bb1 = J[0,0]
        Bb2 = 0
    elif 'xL' in information:
        threshold = ('( x_L = %.2f, x_H=%.2f, L_{10}=%.4f )'
                   % (information['xL'],information['xH'],information['L']/np.log(10)))
        t = 3
        Bb1 = J[0,0]
        Bb2 = J[0,1]
    elif 'L' in information:
        threshold = '( L_{10}=%.4f )' % (information['L']/np.log(10))
        t = 2
        Bb1 = 0
        Bb2 = J[0,0]
    elif 'y0' in information:
        threshold = '( y_0 = %.2f )' % information['y0']
        t = 4
        Bb1 = 0
        Bb2 = J[0,0]
    elif 'u0' in information:
        threshold = '( u_0 = ax+by = %.2f )' % information['u0']
        t = 4
        Bb1 = 0
        Bb2 = J[0,0]
    nc = J.form[1]
    return A)': '%.2f%%' % (C[1,:nc//2].sum() * 100),
            'P(Bb,$0)': '%.2f%%' % (Bb1*100),
            'P(Bb,$100)': '%.2f%%' % (Bb2*100),

exams = ['(T_1)',
         '(T_1 + T_2)',
         '(T_1 rightarrow T_2 )',
         '(T_1 + T_2) (linear)']
print "Exp. value = anticipated value above check T1 ($1)"
print "N(Aa) = anticipated variety of correctly-diagnosed EP-positives per 10M sufferers"
print "N(Ab) = anticipated variety of false negatives per 10M sufferers"
print "N(Ba) = anticipated variety of false positives per 10M sufferers"
print("P(Bb,$0)   = share of sufferers accurately identified EP-negative,n"+ 
      "             no extra value after check T1")
print("P(Bb,$100) = share of sufferers accurately identified EP-negative,n"+ 
      "             check T2 required")
print "P(b|A) = conditional likelihood of a false damaging for EP-positive sufferers"
print "T1 -> T2:   check T1 adopted by check T2 if wanted"
df = pd.DataFrame([present_info(info)
             for info in [info17,info27,info37,info47,info57,
             index=['$%dM, %s' % (cost,test)
                    for cost in [10,100]
                    for check in exams])
colwidths = [12,10,7,7,10,10,10,10,24]
colwidthstyles = [{'selector':'thead th.blank' if i==0
                               else 'th.col_heading.col%d' % (i-1),
             for i,w in enumerate(colwidths)]
df.type.set_table_styles([{'selector':' ','props':
                           {'selector':'thead th',
                            'td span.MathJax_CHTML,td span.MathJax,th span.MathJax_CHTML,th span.MathJax',
                          + colwidthstyles)
Exp. value = anticipated value above check T1 ($1)
N(Aa) = anticipated variety of correctly-diagnosed EP-positives per 10M sufferers
N(Ab) = anticipated variety of false negatives per 10M sufferers
N(Ba) = anticipated variety of false positives per 10M sufferers
P(Bb,$0)   = share of sufferers accurately identified EP-negative,
             no extra value after check T1
P(Bb,$100) = share of sufferers accurately identified EP-negative,
             check T2 required
P(b|A) = conditional likelihood of a false damaging for EP-positive sufferers
T1 -> T2:   check T1 adopted by check T2 if wanted

Exp. value

$10M, (T_1)








( x_0 = 53.16 )

$10M, (T_1 + T_2)








( L_{10}=1.1013 )

$10M, (T_1 rightarrow T_2 )








( x_L = 45.29, x_H=85.99, L_{10}=1.1013 )

$10M, (T_2)








( y_0 = 50.14 )

$10M, (T_1 + T_2) (linear)








( u_0 = ax+by = 50.42 )

$100M, (T_1)








( x_0 = 48.15 )

$100M, (T_1 + T_2)








( L_{10}=0.0973 )

$100M, (T_1 rightarrow T_2 )








( x_L = 40.75, x_H=85.11, L_{10}=0.0973 )

$100M, (T_2)








( y_0 = 47.73 )

$100M, (T_1 + T_2) (linear)








( u_0 = ax+by = 48.22 )

There! We will hold anticipated value down by taking the ( T_1 rightarrow T_2 ) method (check 2 provided that ( x_L le x le x_H )).

With $10M value of a false damaging, we are able to anticipate over 81% of sufferers might be identified accurately as EP-negative with simply the $1 value of the eyeball picture check, and solely 18 sufferers out of ten million (4.5% of EP-positive sufferers) will expertise a false damaging.

With $100M value of a false damaging, we are able to anticipate over 55% of sufferers might be identified accurately as EP-negative with simply the $1 value of the eyeball picture check, and solely 3 sufferers out of ten million (0.7% of EP-positive sufferers) will expertise a false damaging.

And though check ( T_2 ) alone is best than check ( T_1 ) alone, each in retaining anticipated prices down, and in lowering the incidence of each false positives and false negatives, it’s fairly clear that counting on each exams is probably the most engaging possibility.

There’s not an excessive amount of distinction between the mix of exams ( T_1+T_2 ) utilizing ( lambda_{10}(x,y) > L_{10} ) as a quadratic operate of ( x ) and ( y ) primarily based on the logarithm of the likelihood density features, and a linear operate ( u = ax+by > u_0 ) for optimum decisions of ( a,b ). The quadratic operate has a barely decrease anticipated value.

What the @%&^ does this need to do with embedded programs?

We’ve been speaking a lot in regards to the arithmetic of binary classification primarily based on a two-test set of measurements with measurement ( x ) from check ( T_1 ) and measurement ( y ) from check ( T_2 ), the place the likelihood distributions over ( (x,y) ) are a bivariate regular (Gaussian) distribution with barely completely different values of imply ( (mu_x, mu_y) ) and covariance matrix ( start{bmatrix}sigma_x{}^2 & rhosigma_xsigma_y cr rhosigma_xsigma_y & sigma_y{}^2end{bmatrix} ) for 2 circumstances ( A ) and ( B ).

As an embedded system designer, why do you must fear about this?

Properly, you in all probability don’t want the mathematics instantly. It’s price understanding about other ways to make use of the outcomes of the exams.

The extra vital facet is understanding {that a} binary end result primarily based on steady measurements throws away data. For those who measure a sensor voltage and sound an alarm if the sensor voltage ( V > V_0 ) however don’t sound the alarm if ( V le V_0 ) then this doesn’t distinguish the case of ( V ) being near the edge ( V_0 ) from the case the place ( V ) is way away from the edge. We noticed this with fool lights. Take into account letting customers see the unique worth ( V ) as properly, not simply the results of evaluating it with the edge. Or have two thresholds, one indicating a warning and the second indicating an alarm.
This provides the person extra data.

Concentrate on the tradeoffs in selecting a threshold — and that somebody wants to decide on a threshold. For those who decrease the false optimistic price, the false damaging price will improve, and vice versa.

Lastly, though we’ve emphasised the significance of minimizing the possibility of a false damaging (in our fictional embolary pulmonism instance, lacking an EP-positive prognosis could trigger the affected person to die), there are different prices to false positives apart from inflicting pointless prices on customers/sufferers. One that happens within the medical business is “alarm fatigue”, which is mainly a lack of confidence in gear/exams that trigger frequent false positives. It’s the medical system equal to the Aesop’s fable in regards to the boy who cried wolf. One 2011 article said it this fashion:

In a single case, at UMass Memorial Medical Heart in Worcester, nurses failed to answer an alarm that sounded for about 75 minutes, signaling {that a} affected person’s coronary heart monitor battery wanted to get replaced. The battery died, so when the affected person’s coronary heart failed, no disaster alarm sounded.

In one other occasion at Massachusetts Common Hospital final yr, an aged man suffered a deadly coronary heart assault whereas the disaster alarm on his cardiac monitor was turned off and workers didn’t reply to quite a few lower-level alarms warning of a low coronary heart price. Nurses advised state investigators that they had develop into desensitized to the alarms.

“You probably have that many alarms going off on a regular basis, they lose their means to work as an alarm,” Schyve mentioned.

One resolution could also be to desert the ever-present beep and use sounds which were designed to scale back alarm fatigue.

The true takeaway right here is that the human-factors facet of a design (how an alarm is introduced) is commonly as vital or much more vital than the mathematics and engineering behind the design (how an alarm is detected). That is particularly vital in safety-critical conditions comparable to drugs, aviation, nuclear engineering, or industrial equipment.



We’ve talked in nice element as we speak about binary end result exams, the place a number of steady measurements are used to detect the presence of some situation ( A ), whether or not it’s a illness, or an gear failure, or the presence of an intruder.

  • False positives are when the detector erroneously alerts that the situation is current. (penalties: annoyance, misplaced time, pointless prices and use of sources)
  • False negatives are when the detector erroneously doesn’t sign that the situation is current. (penalties: undetected critical situations which will worsen)
  • Alternative of a threshold can commerce off false optimistic and false damaging charges
  • Understanding the bottom price (likelihood that the situation happens) is vital to keep away from the bottom price fallacy
  • If the likelihood distribution of the measurement is a standard (Gaussian) distribution, then an optimum threshold could be chosen to maximise anticipated worth, through the use of the logarithm of the likelihood density operate (PDF), given:

    • the first- and second-order statistics (imply and commonplace deviation for single-variable distributions) of each populations with and with out situation ( A )
    • data of the bottom price
    • estimated prices of all 4 outcomes (true optimistic detection, false optimistic, false damaging, true damaging)

  • A binary end result check could be examined by way of a confusion matrix that reveals chances of all 4 outcomes
  • We will discover the optimum threshold by understanding that if a measurement happens at the optimum threshold, each optimistic and damaging diagnoses have the identical anticipated worth, primarily based the conditional likelihood of prevalence given the measured worth
  • Bayes’ Rule can be utilized to compute the conditional likelihood of ( A ) given the measured worth, from its “converse”, the conditional likelihood of the measured worth with and with out situation ( A )
  • Fool lights, the place the detected binary end result is introduced as an alternative of the unique measurement, disguise data from the person, and must be used with warning
  • The distinguishability or separation coefficient ( d’ ) can be utilized to characterize how far aside two likelihood distributions are, basically measuring the distinction of the means, divided by an efficient commonplace deviation
  • If two exams are doable, with a cheap check ( T_1 ) providing a mediocre worth of ( d’ ), and a dearer check ( T_2 ) providing a better worth of ( d’ ), then the 2 exams could be mixed to assist scale back general anticipated prices. We explored one state of affairs (the fictional “embolary pulmonism” or EP) the place the likelihood distribution was a recognized bivariate regular distribution. 5 strategies have been used, so as of accelerating effectiveness:

    • Take a look at ( T_1 ) solely, producing a measurement ( x ), detecting situation ( A ) if ( x>x_0 ) (highest anticipated value)
    • Take a look at ( T_2 ) solely, producing a measurement ( y ), detecting situation ( A ) if ( y>y_0 )
    • Take a look at ( T_1 ) and ( T_2 ), mixed by a linear mixture ( u = ax+by ), detecting situation ( A ) if ( u>u_0 ), with ( a ) and ( b ) chosen to maximise ( d’ )
    • Take a look at ( T_1 ) and ( T_2 ), mixed by a quadratic operate ( lambda(x,y) = a(x-mu_x)^2 + b(x-mu_x)(y-mu_y) + c(y-mu_y)^2 ) primarily based on the log of the ratio of PDFs, detecting situation ( A ) if ( lambda(x,y) > L ) (in our instance, this has a barely decrease value than the linear mixture)
    • Take a look at ( T_1 ) used to detect three circumstances primarily based on thresholds ( x_L ) and ( x_H ) (lowest anticipated value):

      • Diagnose absense of situation ( A ) if ( x < x_L ), no additional check vital
      • Diagnose presence of situation ( A ) if ( x > x_H ), no additional check vital
      • If ( x_L le x le x_H ), carry out check ( T_2 ) and diagnose primarily based on the measurements from each exams (we explored utilizing the quadratic operate on this article)

  • An extreme false optimistic price could cause “alarm fatigue” through which true positives could also be missed due to an inclination to disregard detected situations

Whew! OK, that’s all for now.

© 2019 Jason M. Sachs, all rights reserved.

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