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# A patient was seen for pain, tenderness, and edema of the right wrist. She experienced a...

###### Question:

A patient was seen for pain, tenderness, and edema of the right wrist. She experienced a low-impact injury when she braced herself against the wall after she slipped without falling from a step stool in the kitchen pantry of her home. Scans indicate there is a fracture due to osteoporosis and weakening of bone.

Final diagnosis and procedure: (1) Postmenopausal osteoporotic fracture of distal radial extra-articular metaphysis, (2) open reduction with internal fixation using locking-plate system.

Assign the appropriate diagnosis & procedure codes.ICD-10-CM and ICD-10-PCS

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Here's the question you clicked on:

jennychan12 Group Title Integrals (attached) one year ago one year ago

• This Question is Closed
1. jennychan12 Group Title

true or false? $\int\limits_{a}^{b} [f(x)g(x)dx] = [\int\limits_{a}^{b} [f(x)]dx][\int\limits_{a}^{b}[g(x)dx]$ i wanna say true because there's a limit definition of the same kind of thing...

2. sirm3d Group Title

that's false. take f(x) = g(x) =1, a = 0, b = 1/2


text:
Development¶

Basics and Commonalities¶

Before stepping into fitting development patterns, its worth reviewing the basics of Estimators. The main modeling API implemented by chainladder follows that of the scikit-learn estimator. An estimator is any object that learns from data.

Scikit-Learn API¶

The scikit-learn API is a common modeling interface that is used to construct and fit a countless variety of machine learning algorithms. The common interface allows for very quick swapping between models with minimal code changes. The chainladder package has adopted the interface to promote a standardized approach to fitting reserving models.

All estimator objects can optionally be configured with parameters to uniquely specify the model being built. This is done ahead of pushing any data through the model.

estimator = Estimator(param1=1, param2=2)

All estimator objects expose a fit method that takes a Triangle as input, X:

estimator.fit(X=data)

All estimators include a sample_weight option to the fit method to specify an exposure basis. If an exposure base is not applicable, then this argument is ignored.

estimator.fit(X=data, sample_weight=weight)

All estimators either transform the input Triangle or predict an outcome.

Transformers¶

All transformers include a transform method. The method is used to transform a Triangle and it will always return a Triangle with added features based on the specifics of the transformer.

transformed_data = estimator.transform(data)

Other than final IBNR models, chainladder estimators are transformers. That is, they return your Triangle back to you with additional properties.

Transforming can be done at the time of fit.

# Fitting and Transforming
estimator.fit(data)
transformed_data = estimator.transform(data)
# One line equivalent
transformed_data = estimator.fit_transform(data)
assert isinstance(transformed_data, cl.Triangle)

Predictors¶

All predictors include a predict method.

prediction = estimator.predict(new_data)

Predictors are intended to create new predictions. It is not uncommon to fit a model on a more aggregate view, say national level, of data and predict on a more granular triangle, state or provincial.

Parameter Types¶

Estimator parameters: All the parameters of an estimator can be set when it is instantiated or by modifying the corresponding attribute. These parameters define how you’d like to fit an estimator and are chosen before the fitting process. These are often referred to as hyperparameters in the context of Machine Learning, and throughout these documents. Most of the hyperparameters of the chainladder package take on sensible defaults.

estimator = Estimator(param1=1, param2=2)
assert estimator.param1 == 1

Estimated parameters: When data is fitted with an estimator, parameters are estimated from the data at hand. All the estimated parameters are attributes of the estimator object ending by an underscore. The use of the underscore is a key API design style of scikit-learn that allows for the quicker recognition of fitted parameters vs hyperparameters:

estimator.estimated_param_

In many cases the estimated parameters are themselves Triangles and can be manipulated using the same methods we learned about in the Triangle class.

import chainladder as cl
import pandas as pd
import matplotlib.pyplot as plt
plt.style.use('ggplot')
%config InlineBackend.figure_format = 'svg'

Matplotlib is building the font cache; this may take a moment.

dev = cl.Development().fit(cl.load_sample('ukmotor'))
type(dev.cdf_)

chainladder.core.triangle.Triangle

Commonalities¶

All “Development Estimators” are transformers and reveal common a set of properties when they are fit.

1. ldf_ represents the fitted age-to-age factors of the model.

2. cdf_ represents the fitted age-to-ultimate factors of the model.

3. All “Development estimators” implement the transform method.

cdf_ is nothing more than the cumulative representation of the ldf_ vectors.

dev = cl.Development().fit(cl.load_sample('raa'))
dev.ldf_.incr_to_cum() == dev.cdf_

True

Development¶

Development allows for the selection of loss development patterns. Many of the typical averaging techniques are available in this class: simple, volume and regression through the origin. Additionally, Development includes patterns to allow for fine-tuned exclusion of link-ratios from the LDF calculation.

raa = cl.load_sample('raa')
cl.Development(average='simple')

Development(average='simple')

Alternatively, you can provide a list to parameterize each development period separately. When adjusting individual development periods the list must be the same length as your triangles link_ratio development axis.

len(raa.link_ratio.development)

9

cl.Development(average=['volume']+['simple']*8)

Development(average=['volume', 'simple', 'simple', 'simple', 'simple', 'simple',
'simple', 'simple', 'simple'])

This approach works for average, n_periods, drop_high and drop_low.

Notice where you have not specified a parameter, a sensible default is chosen for you.

Transforming¶

When transforming a Triangle, you will receive a copy of the original triangle back along with the fitted properties of the Development estimator. Where the original Triangle contains all link ratios, the transformed version recognizes any ommissions you specify.

triangle = cl.load_sample('raa')
dev = cl.Development(drop=('1982', 12), drop_valuation='1988')
transformed_triangle = dev.fit_transform(triangle)
transformed_triangle.ldf_

12-24 24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120
(All) 2.6625 1.5447 1.2975 1.1719 1.1134 1.0468 1.0294 1.0331 1.0092
transformed_triangle.link_ratio.heatmap()

12-24 24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120
1981 1.6498 1.3190 1.0823 1.1469 1.1951 1.1130 1.0333 1.0092
1982 1.2593 1.9766 1.2921 1.1318 0.9934 1.0331
1983 2.6370 1.5428 1.1635 1.1607 1.1857 1.0264
1984 2.0433 1.3644 1.3489 1.1015 1.0377
1985 8.7592 1.6556 1.3999 1.0087
1986 4.2597 1.8157 1.2255
1987 7.2172 1.1250
1988 1.8874
1989 1.7220

By decoupling the fit and transform methods, we can apply our Development estimator to new data. This is a common pattern of the scikit-learn API. In this example we generate development patterns at an industry level and apply those patterns to individual companies.

clrd = cl.load_sample('clrd')
clrd = clrd[clrd['LOB']=='wkcomp']['CumPaidLoss']
# Summarize Triangle to industry level to estimate patterns
dev = cl.Development().fit(clrd.sum())
# Apply Industry patterns to individual companies
dev.transform(clrd)

Triangle Summary
Valuation: 1997-12
Grain: OYDY
Shape: (132, 1, 10, 10)
Index: [GRNAME, LOB]
Columns: [CumPaidLoss]

Groupby¶

Triangles have a groupby method that follows pandas syntax and this allows for aggregating triangle data to a more reasonable level for any particular analysis. However, it is often the desire of an actuary to estimate development factors at a more aggregate grain generally and then apply it to a more detailed triangle.

We can, for example, pick volume-weighted development patterns at a Line of Business level and subsequently apply them to each company within the line of business as follows:

clrd = cl.load_sample('clrd')['CumPaidLoss']
clrd = cl.Development(groupby='LOB').fit_transform(clrd)

clrd.shape, clrd.ldf_.shape

((775, 1, 10, 10), (6, 1, 1, 9))

Notice we’ve retained the grain of the original triangle, but there are six sets of development patterns, one for each line of business. Using this transformed triangle in an IBNR esimtator will result in IBNR at the original grain but using patterns at the Line of Business grain.

It is worth noting that fitting and transforming are entirely decoupled from one another, and we could achieve the same outcome by directly aggregating the Triangle before passing to the fit method.

clrd = cl.load_sample('clrd')['CumPaidLoss']
model = cl.Development().fit(clrd.groupby('LOB').sum())
clrd = model.transform(clrd)

clrd.shape, clrd.ldf_.shape

((775, 1, 10, 10), (6, 1, 1, 9))

This begs the question, why do we need a groupby hyperparameter as part of the Development estimator when we can aggregate the Triangle before fitting? In more advanced situations, we will be creating compound estimators called Pipelines which are very powerful for building custom workflows, but with the limitation that fitting and transforming have to be coupled together. You can explore this in more detail in the Pipeline section.

DevelopmentConstant¶

The DevelopmentConstant estimator simply allows you to hard code development patterns into a Development Estimator. A common example would be to include a set of industry development patterns in your workflow that are not directly estimated from any of your own data.

triangle = cl.load_sample('ukmotor')
patterns={12: 2, 24: 1.25, 36: 1.1, 48: 1.08, 60: 1.05, 72: 1.02}
cl.DevelopmentConstant(patterns=patterns, style='ldf').fit(triangle).ldf_

12-24 24-36 36-48 48-60 60-72 72-84
(All) 2.0000 1.2500 1.1000 1.0800 1.0500 1.0200

By wrapping patterns in the DevelopmentConstant estimator, we can integrate into a larger workflow with tail extrapolation and IBNR calculations.

Examples¶

The IncrementalAdditive method uses both the triangle of incremental losses and the exposure vector for each accident year as a base. Incremental additive ratios are computed by taking the ratio of incremental loss to the exposure (which has been adjusted for the measurable effect of inflation), for each accident year. This gives the amount of incremental loss in each year and at each age expressed as a percentage of exposure, which we then use to square the incremental triangle.

tri = cl.load_sample("ia_sample")
X=tri['loss'],
sample_weight=tri['exposure'].latest_diagonal)
ia.incremental_.round(0)

12 24 36 48 60 72
2000 1,001.00 854.00 568.00 565.00 347.00 148.00
2001 1,113.00 990.00 671.00 648.00 422.00 164.00
2002 1,265.00 1,168.00 800.00 744.00 482.00 195.00
2003 1,490.00 1,383.00 1,007.00 849.00 543.00 220.00
2004 1,725.00 1,536.00 1,068.00 984.00 629.00 255.00
2005 1,889.00 1,811.00 1,256.00 1,157.00 740.00 300.00

These incremental_ values are then used to determine an implied set of mutiplicative development patterns. Because incremental additive values are unique for each origin, so too will be the ldf_.

ia.ldf_

12-24 24-36 36-48 48-60 60-72
2000 1.8531 1.3062 1.2332 1.1161 1.0444
2001 1.8895 1.3191 1.2336 1.1233 1.0426
2002 1.9233 1.3288 1.2301 1.1212 1.0438
2003 1.9282 1.3505 1.2188 1.1148 1.0418
2004 1.8904 1.3276 1.2274 1.1184 1.0429
2005 1.9586 1.3395 1.2335 1.1210 1.0438

Incremental calculation¶

The estimation of the incremental triangle can be done with varying hyperparameters of n_period and average similar to the Development estimator. Additionally, a trend in the origin period can also be selected.

Suppose there is a vector zeta_ that represents an estimate of the incremental losses, X for a development period as a percentage of some exposure or sample_weight. Using a ‘volume’ weighted estimate for all origin periods, we can manually estimate zeta_.

zeta_ = tri['loss'].cum_to_incr().sum('origin') / tri['exposure'].sum('origin')
zeta_

12 24 36 48 60 72
2000 0.2432 0.2220 0.1540 0.1419 0.0907 0.0368

The zeta_ vector along with the sample_weight and optionally a trend are used to propagate incremental losses to the lower half of the Triangle. In the trivial case of no trend, we can estimate the incrementals for age 72.

zeta_.loc[..., 72] * tri['exposure'].latest_diagonal

72
2000 148.00
2001 163.85
2002 195.43
2003 220.11
2004 255.15
2005 299.97

These are the same incrementals that the IncrementalAdditive method produces.

zeta_.loc[..., 72]*tri['exposure'].latest_diagonal == ia.incremental_.loc[..., 72]

True

The MunichAdjustment is a bivariate adjustment to loss development factors. There is a fundamental correlation between the paid and the case incurred data of a triangle. The ratio of paid to incurred (P/I) has information that can be used to simultaneously adjust the basic development factor selections for the two separate triangles.

Depending on whether the momentary (P/I) ratio is below or above average, one should use an above-average or below-average paid development factor and/or a below-average or above-average incurred development factor. In doing so, the model replaces a set of development patterns that would be used for all origins with individualized development curves that reflect the unique levels of (P/I) per origin period.

BerquistSherman Comparison¶

This method is similar to the BerquistSherman approach in that it tries to adjust for case reserve adequacy. However it is different in two distinct ways.

1. The BerquistSherman method is a direct adjustment to the data whereas the MunichAdjustment keeps the Triangle intact and adjusts the development patterns.

2. The MunichAdjustment is built in the context of a stochastic framework.

Residuals¶

The MunichAdjustment uses the correlation between the residuals of the univariate (basic) model and the (P/I) model. These correlations spin off a property lambda_ which is represented by the line through the origin of the correlation plots.

With the correlations, lambda_ known, the basic development patterns can be adjusted based on the (P/I) ratio at any given cell of the Triangle.

[QM04]

ClarkLDF¶

ClarkLDF estimates growth curves of the form ‘loglogistic’ or ‘weibull’ for the incremental loss development of a Triangle. These growth curves are monotonic increasing and are more relevant for paid data. While the model can be used for case incurred data, if there is too much “negative” development, other Estimators should be used.

The Loglogistic Growth Function:

$$G(x|\omega, \theta) =\frac{x^{\omega }}{x^{\omega } + \theta^{\omega }}$$

The Weibull Growth Function:

$$G(x|\omega, \theta) =1-exp(-\left (\frac{x}{\theta} \right )^\omega)$$

Parameterized growth curves can produce patterns for any age and can even be used to estimate a tail beyond the latest age in a Triangle. In general, the loglogistic growth curve produces a larger tail than the weibull growth curve.

LDF and Cape Cod methods¶

Clark approaches curve fitting with two different methods, an LDF approach and a Cape Cod approach. The LDF approach only requires a loss triangle whereas the Cape Cod approach would also need a premium vector. Choosing between the two methods occurs at the time you fit the estimator. When a premium vector is included, the Cape Cod method is invoked.

A simple example of using ClarkLDF LDF Method. Upon fitting the Estimator, we obtain both omega_ and theta_.

clrd = cl.load_sample('clrd').groupby('LOB').sum()
dev = cl.ClarkLDF(growth='weibull').fit(clrd['CumPaidLoss'])
dev.omega_

CumPaidLoss
LOB
comauto 0.928924
medmal 1.569651
othliab 1.330078
ppauto 0.831528
prodliab 1.456207
wkcomp 0.898285

Perhaps more useful than the parameters is the growth curve G_ function they represent which can be used to deetermine the development factor at any age.

1/dev.G_(37.5).to_frame()

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/conda/latest/lib/python3.10/site-packages/chainladder/core/pandas.py:62: UserWarning: In an upcoming version of the package, origin_as_datetime will be defaulted to True in to_frame(...), use origin_as_datetime=False to preserve current setting.
warnings.warn(warning)

LOB
comauto     1.270910
medmal      1.707220
othliab     1.619184
ppauto      1.118805
prodliab    2.126036
wkcomp      1.311571
dtype: float64

Another example showing the usage of the ClarkLDF Cape Cod approach. With the Cape Cod, an Expected Loss Ratio is included as an extra feature in the elr_ property.

cl.ClarkLDF().fit(
X=clrd['CumPaidLoss'],
sample_weight=clrd['EarnedPremDIR'].latest_diagonal
).elr_

CumPaidLoss
LOB
comauto 0.680305
medmal 0.701454
othliab 0.623769
ppauto 0.825925
prodliab 0.671075
wkcomp 0.697938

Residuals¶

Clark’s model assumes Incremental losses are independent and identically distributed. To ensure compatibility with this assumption, he suggests reviewing the “Normalized Residuals” of the fitted incremental losses to ensure the assumption is not violated.

Stochastics¶

Using MLE to solve for the growth curves, we can produce statistics about the parameter and process uncertainty of our model.

[Cla03]

CaseOutstanding¶

The CaseOutstanding method is a deterministic method that estimates incremental payment patterns from prior lag carried case reserves. Included in this is also patterns for the carried case reserves based on the prior lag carried case reserve.

Like the MunichAdjustment and BerquistSherman, this estimator is useful when you want to incorporate information about case reserves into paid ultimates.

To use it, a triangle with both paid and incurred amounts must be available.

tri = cl.load_sample('usauto')
model = cl.CaseOutstanding(paid_to_incurred=('paid', 'incurred')).fit(tri)
model.paid_ldf_

=== in _set_ldf ===
=== self.case ===
12            24            36            48            60             72             84             96             108            120
1998  1.798098e+07  9.602188e+06  5.414189e+06  2.867316e+06  1.405073e+06  722000.560555  397562.100227  251150.266276  169222.000000   98117.000000
1999  1.968226e+07  1.051070e+07  5.926455e+06  3.138608e+06  1.538014e+06  790312.901593  435177.580406  274912.938466  185233.000000  107400.375016
2000  2.181513e+07  1.164970e+07  6.568678e+06  3.478725e+06  1.704682e+06  875955.506899  482335.790378  304704.000000  205305.855545  119038.863909
2001  1.904741e+07  1.017168e+07  5.735298e+06  3.037373e+06  1.488406e+06  764821.490526  421141.000000  266045.667404  179258.340423  103936.193801
2002  1.956232e+07  1.044665e+07  5.890341e+06  3.119483e+06  1.528642e+06  785497.000000  432525.754277  273237.711280  184104.252262  106745.913175
2003  2.091944e+07  1.117138e+07  6.298978e+06  3.335894e+06  1.634690e+06  839990.148108  462531.839581  292193.331822  196876.319229  114151.314922
2004  2.007969e+07  1.072294e+07  6.046125e+06  3.201985e+06  1.569070e+06  806271.365800  443964.942762  280464.142653  188973.333980  109569.066729
2005  2.039616e+07  1.089194e+07  6.141416e+06  3.252450e+06  1.593800e+06  818978.706486  450962.107761  284884.432846  191951.671841  111295.943707
2006  2.066376e+07  1.103484e+07  6.221991e+06  3.295122e+06  1.614710e+06  829723.593216  456878.667897  288622.076985  194470.051080  112756.131011
2007  2.162359e+07  1.154741e+07  6.511004e+06  3.448181e+06  1.689714e+06  868264.503103  478100.819120  302028.659070  203503.243309  117993.687132
=== self.paid ===
12            24            36            48            60            72            84            96            108           120
1998  18539254.0  3.323104e+07  4.006201e+07  4.389204e+07  4.589654e+07  4.676542e+07  4.722132e+07  4.744688e+07  4.755546e+07  4.764419e+07
1999  20410193.0  3.609068e+07  4.325940e+07  4.715924e+07  4.920853e+07  5.016204e+07  5.062576e+07  5.087881e+07  5.100053e+07  5.109766e+07
2000  22120843.0  3.897601e+07  4.638928e+07  5.056238e+07  5.273528e+07  5.374010e+07  5.428433e+07  5.453322e+07  5.466649e+07  5.477415e+07
2001  22992259.0  4.009620e+07  4.776784e+07  5.209392e+07  5.436344e+07  5.537880e+07  5.587842e+07  5.611147e+07  5.622784e+07  5.632183e+07
2002  24092782.0  4.179531e+07  4.990380e+07  5.435288e+07  5.675438e+07  5.780722e+07  5.829581e+07  5.853516e+07  5.865467e+07  5.875120e+07
2003  24084451.0  4.139961e+07  4.907033e+07  5.358420e+07  5.593065e+07  5.697291e+07  5.749540e+07  5.775136e+07  5.787916e+07  5.798239e+07
2004  24369770.0  4.148986e+07  4.923668e+07  5.377467e+07  5.600590e+07  5.700632e+07  5.750783e+07  5.775352e+07  5.787618e+07  5.797527e+07
2005  25100697.0  4.270223e+07  5.064499e+07  5.499527e+07  5.726166e+07  5.827785e+07  5.878727e+07  5.903682e+07  5.916142e+07  5.926207e+07
2006  25608776.0  4.360650e+07  5.144103e+07  5.584838e+07  5.814450e+07  5.917403e+07  5.969013e+07  5.994296e+07  6.006919e+07  6.017116e+07
2007  27229969.0  4.545463e+07  5.365308e+07  5.826515e+07  6.066793e+07  6.174527e+07  6.228535e+07  6.254992e+07  6.268202e+07  6.278873e+07
=== set LDF return ===
Triangle Summary
Valuation:           2261-12
Grain:                  OYDY
Shape:         (1, 2, 10, 9)
Index:               [Total]
Columns:    [incurred, paid]

24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120 120-132
(All) 0.8428 0.7100 0.7084 0.6968 0.6376 0.6220 0.5534 0.4374 0.5243

In the example above, the incremental paid losses during the period 12-24 is expected to be 84.28% of the outstanding case reserve at lag 12. The set of patterns produced by CaseOutstanding don’t follow the multiplicative approach commonly used in the various IBNR methods making them not directly usable. Because of this, the estimator determines the ‘implied’ multiplicative pattern so that a broader set of IBNR methods can be used. Due to the origin period specifics on case reserves, each origin gets its own set of multiplicative ldf_ patterns.

model.ldf_['paid']

12-24 24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120
1998 1.7925 1.2056 1.0956 1.0457 1.0189 1.0097 1.0048 1.0023 1.0019
1999 1.7683 1.1986 1.0902 1.0435 1.0194 1.0092 1.0050 1.0024 1.0019
2000 1.7620 1.1902 1.0900 1.0430 1.0191 1.0101 1.0046 1.0024 1.0020
2001 1.7439 1.1913 1.0906 1.0436 1.0187 1.0090 1.0042 1.0021 1.0017
2002 1.7348 1.1940 1.0892 1.0442 1.0186 1.0085 1.0041 1.0020 1.0016
2003 1.7189 1.1853 1.0920 1.0438 1.0186 1.0092 1.0045 1.0022 1.0018
2004 1.7025 1.1867 1.0922 1.0415 1.0179 1.0088 1.0043 1.0021 1.0017
2005 1.7012 1.1860 1.0859 1.0412 1.0177 1.0087 1.0042 1.0021 1.0017
2006 1.7028 1.1797 1.0857 1.0411 1.0177 1.0087 1.0042 1.0021 1.0017
2007 1.6693 1.1804 1.0860 1.0412 1.0178 1.0087 1.0042 1.0021 1.0017

Incremental patterns¶

The incremental patterns of the CaseOutstanding method are avilable as additional properties for review. They are the paid_to_prior_case_ and the case_to_prior_case_. These are useful to review when deciding on the appropriate hyperparameters for paid_n_periods and case_n_periods. Once you are satisfied with your hyperparameter tuning, you can see the fitted selections in the paid_ldf_ and case_ldf_ incremental patterns.

model.case_to_prior_case_

24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120 120-132
1998 0.5378 0.5541 0.5253 0.4981 0.5329 0.5380 0.5877 0.6970 0.5798
1999 0.5368 0.5649 0.5442 0.4969 0.5029 0.5800 0.6420 0.6506
2000 0.5461 0.5742 0.5391 0.4872 0.5376 0.5432 0.6655
2001 0.5406 0.5660 0.5148 0.5013 0.5077 0.5414
2002 0.5409 0.5546 0.5406 0.4802 0.4881
2003 0.5265 0.5765 0.5363 0.4764
2004 0.5298 0.5665 0.5069
2005 0.5215 0.5539
2006 0.5261
2007
model.case_ldf_

24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120 120-132
(All) 0.5340 0.5638 0.5296 0.4900 0.5139 0.5506 0.6317 0.6738 0.5798

[Fri10]

TweedieGLM¶

The TweedieGLM implements the GLM reserving structure discussed by Taylor and McGuire. A nice property of the GLM framework is that it is highly flexible in terms of including covariates that may be predictive of loss reserves while maintaining a close relationship to traditional methods. Additionally, the framework can be extended in a straightforward way to incorporate various approaches to measuring prediction errors. Behind the scenes, TweedieGLM is using scikit-learn’s TweedieRegressor estimator.

Long Format¶

GLMs are fit to triangles in “Long Format”. That is, they are converted to pandas DataFrames behind the scenes. Each axis of the Triangle is included in the dataframe. The origin and development axes are in columns of the same name. You can inspect what your Triangle looks like in long format by calling to_frame with keepdims=True

cl.load_sample('clrd').to_frame(keepdims=True).reset_index().head()

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/conda/latest/lib/python3.10/site-packages/chainladder/core/pandas.py:62: UserWarning: In an upcoming version of the package, origin_as_datetime will be defaulted to True in to_frame(...), use origin_as_datetime=False to preserve current setting.
warnings.warn(warning)

GRNAME LOB origin development IncurLoss CumPaidLoss BulkLoss EarnedPremDIR EarnedPremCeded EarnedPremNet
0 Adriatic Ins Co othliab 1995 12 8.0 NaN 8.0 139.0 131.0 8.0
1 Adriatic Ins Co othliab 1995 24 11.0 NaN 4.0 139.0 131.0 8.0
2 Adriatic Ins Co othliab 1995 36 7.0 3.0 4.0 139.0 131.0 8.0
3 Adriatic Ins Co othliab 1996 12 40.0 NaN 40.0 410.0 359.0 51.0
4 Adriatic Ins Co othliab 1996 24 40.0 NaN 40.0 410.0 359.0 51.0

Warning

‘origin’, ‘development’, and ‘valuation’ are reserved keywords for the dataframe. Declaring columns with these names separately will result in error.

While you can inspect the Triangle in long format, you will not directly convert to long format yourself. The TweedieGLM does this for you. Additionally, the origin of the design matrix is restated in years from the earliest origin period. That is, is if the earliest origin is ‘1995-01-01’ then it gets replaced with 0. Consequently, ‘1996-04-01’ would be replaced with 1.25. This is done because datetimes have limited support in scikit-learn. Finally, the TweedieGLM will automatically convert the response to an incremental basis.

R-style formulas¶

We use the patsy library to allow formulation of the the feature set X of the GLM. Because X is a parameter that used extensively throughout chainladder, the TweedieGLM refers to it as the design_matrix. Those familiar with the R programming language will be familiar with the notation used by patsy. For example, we can include both origin and development as terms in a model.

genins = cl.load_sample('genins')
glm = cl.TweedieGLM(design_matrix='development + origin').fit(genins)
glm.coef_

coef_
Intercept 13.516322
development -0.006251
origin 0.033863

Replicating the results of the volume weighted chainladder development patterns can be done by fitting a Poisson-log GLM to incremental paids. To do this, we can specify the power and link of the estimator as well as the design_matrix. The volume-weighted chainladder method can be replicated by including both origin and development as categorical features.

dev = cl.TweedieGLM(
design_matrix='C(development) + C(origin)',

A trivial comparison against the traditional Development estimator shows a comparable set of ldf_ patterns.

Parsimonious modeling¶

Having full access to all axes of the Triangle along with the powerful formulation of patsy allows for substantial customization of the model fit. For example, we can include ‘LOB’ interactions with piecewise linear coefficients to reduce model complexity.

clrd = cl.load_sample('clrd')['CumPaidLoss'].groupby('LOB').sum()
clrd=clrd[clrd['LOB'].isin(['ppauto', 'comauto'])]
dev = cl.TweedieGLM(
design_matrix='LOB+LOB:C(np.minimum(development, 36))+LOB:development+LOB:origin',
max_iter=1000).fit(clrd)
dev.coef_

coef_
Intercept 12.549946
LOB[T.ppauto] 3.202703
LOB[comauto]:C(np.minimum(development, 36))[T.24] 0.578694
LOB[ppauto]:C(np.minimum(development, 36))[T.24] 0.449833
LOB[comauto]:C(np.minimum(development, 36))[T.36] 0.790514
LOB[ppauto]:C(np.minimum(development, 36))[T.36] 0.321208
LOB[comauto]:development -0.044627
LOB[ppauto]:development -0.054814
LOB[comauto]:origin 0.054581
LOB[ppauto]:origin 0.057790

This model is limited to 10 coefficients across two lines of business. The basic chainladder model is known to be overparameterized with at least 18 parameters requiring estimation. Despite drastically simplifying the model, the cdf_ patterns of the GLM are within 1% of the traditional chainladder for every lag and for both lines of business:

((dev.cdf_.iloc[..., 0, :] /
cl.Development().fit(clrd).cdf_) - 1
).to_frame().round(3)

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/conda/latest/lib/python3.10/site-packages/chainladder/core/pandas.py:62: UserWarning: In an upcoming version of the package, origin_as_datetime will be defaulted to True in to_frame(...), use origin_as_datetime=False to preserve current setting.
warnings.warn(warning)

development 12-Ult 24-Ult 36-Ult 48-Ult 60-Ult 72-Ult 84-Ult 96-Ult 108-Ult
LOB
comauto 0.002 0.003 -0.01 0.003 0.011 0.008 0.005 -0.000 -0.002
ppauto 0.006 0.003 -0.00 0.001 0.002 0.001 0.001 0.001 0.001

Like every other Development estimator, the TweedieGLM produces a set of ldf_ patterns and can be used in a larger workflow with tail extrapolation and reserve estimation.

[TG16]

DevelopmentML¶

DevelopmentML is a general development estimator that works as an interface to scikit-learn compliant machine learning (ML) estimators. The TweedieGLM is a special case of DevelopmentML with the ML algorithm limited to scikit-learn’s TweedieRegressor estimator.

The Interface¶

ML algorithms are designed to be fit against tabular data like a pandas DataFrame or a 2D numpy array. A Triangle does not meet the definition and so DevelopmentML is provided to incorporate ML into a broader reserving workflow. This includes:

1. Automatic conversion of Triangle to a dataframe for fitting

2. Flexibility in expressing any preprocessing as part of a scikit-learn Pipeline

3. Predictions through the terminal development age of a Triangle to fill in the lower half

4. Predictions converted to ldf_ patterns so that the results of the estimator are compliant with the rest of chainladder, like tail selection and IBNR modeling.

Features¶

Data from any axis of a Triangle is available to be used in the DevelopmentML estimator. For example, we can use many of the scikit-learn components to generate development patterns from both the time axes as well as the index of the Triangle.

from sklearn.ensemble import RandomForestRegressor
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import OneHotEncoder
from sklearn.compose import ColumnTransformer

# Decide how to preprocess the X (ML) dataset using sklearn
design_matrix = ColumnTransformer(transformers=[
('dummy', OneHotEncoder(drop='first'), ['LOB', 'development']),
('passthrough', 'passthrough', ['origin'])
])

# Wrap preprocessing and model in a larger sklearn Pipeline
estimator_ml = Pipeline(steps=[
('design_matrix', design_matrix),
('model', RandomForestRegressor())
])

# Fitting DevelopmentML fits the underlying ML model and gives access to ldf_
cl.DevelopmentML(estimator_ml=estimator_ml, y_ml='CumPaidLoss').fit(clrd).ldf_

Triangle Summary
Valuation: 2261-12
Grain: OYDY
Shape: (6, 1, 10, 9)
Index: [LOB]
Columns: [CumPaidLoss]

Autoregressive¶

The time-series nature of loss development naturally lends to an urge for autoregressive features. That is, features that are based on predictions, albeit on a lagged basis. DevelopmentML includes an autoregressive parameter that can be used to express the response as a lagged feature as well.

Note

When using autoregressive features, you must also declare it as a column in your estimator_ml Pipeline.

PatsyFormula¶

While the sklearn preprocessing API is powerful, it can be tedious work with in some instances. In particular, modeling complex interactions is much easier to do with Patsy. The chainladder package includes a custom sklearn estimator to gain access to the patsy API. This is done through the PatsyFormula estimator.

estimator_ml = Pipeline(steps=[
('design_matrix', cl.PatsyFormula('LOB:C(origin)+LOB:C(development)+development')),
('model', RandomForestRegressor())
])
cl.DevelopmentML(
estimator_ml=estimator_ml,
y_ml='CumPaidLoss').fit(clrd).ldf_.iloc[0, 0, 0].round(2)

12-24 24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120
(All) 2.4800 1.4100 1.1900 1.0900 1.0400 1.0200 1.0100 1.0100 1.0100

Note

PatsyFormula is not an estimator designed to work with triangles. It is an sklearn transformer designed to work with pandas DataFrames allowing it to work directly in an sklearn Pipeline.

BarnettZehnwirth¶

The BarnettZehnwirth estimator solves for development patterns using the Probabilistic Trend Family (PTF) regression framework. Unlike the ELRF framework, which assumes no valuation covariate, the PTF framework allows for this.

Structurally, the PTF regression is different from the ELRF (ELRF) regression framework in two distinct ways:

1. Where the ELRF fits independent regressions to each adjacent development lag, the PTF regression is fit to the entire triangle

2. Where the ELRF is fit to cumulative amounts, the PTF is fit to the log of the incremental amounts of the Triangle.

Formulation¶

The PTF framework is an ordinary least squares (OLS) model with the response, y being the log of the incremental amounts of a Triangle. These are assumed to be normally distributed which implies the incrementals themselves are log-normal distributed.

The framework includes coefficients for origin periods (alpha), development periods (gamma) and calendar period (iota).

$$y(i, j) = \alpha _{i} + \sum_{k=1}^{j}\gamma _{k}+ \sum_{\iota =1}^{i+j}\gamma _{\iota}+ \varepsilon _{i,j}$$

These coefficients can be categorical or continuous, and to support a wide range of model forms, patsy formulas are used.

abc = cl.load_sample('abc')

# Discrete origin, development, valuation
cl.BarnettZehnwirth(formula='C(origin)+C(development)').fit(abc).coef_.T

Intercept C(origin)[T.1.0] C(origin)[T.2.0] C(origin)[T.3.0] C(origin)[T.4.0] C(origin)[T.5.0] C(origin)[T.6.0] C(origin)[T.7.0] C(origin)[T.8.0] C(origin)[T.9.0] ... C(development)[T.24] C(development)[T.36] C(development)[T.48] C(development)[T.60] C(development)[T.72] C(development)[T.84] C(development)[T.96] C(development)[T.108] C(development)[T.120] C(development)[T.132]
coef_ 11.836863 0.178824 0.345112 0.378133 0.405093 0.427041 0.431076 0.660383 0.963223 1.1568 ... 0.251091 -0.055824 -0.448589 -0.828917 -1.16913 -1.507561 -1.798345 -2.0231 -2.238333 -2.427672

1 rows × 21 columns

# Linear coefficients for origin, development, and valuation
cl.BarnettZehnwirth(formula='origin+development+valuation').fit(abc).coef_.T

Intercept origin development valuation
coef_ 8.359157 4.215981 0.319288 -4.116569

The PTF framework is particularly useful when there is calendar period inflation influences on loss development.

[BZ00]


text:
# Density of States

1. Aug 16, 2010

### xGAME-OVERx

Dear all,

I am doing summer research in the field of Molecular Quantum Electro Dynamics, and a persistent problem that I (and occasionally the group I am working with) have/has is knowing what to use as the density of final states (radiation states).

It is required in the Fermi Golden Rule, when calculating a rate:

$$\Gamma={{2\pi}\over{\hbar}}|M_{fi}|^2\rho$$

The most accepted definition in all the textbooks I can find is:

$$\rho={{k^2d\Omega}\over{(2\pi)^3{\hbar}c}}V$$

Which is useful, except for in two circumstances:
1) When there is not a well defined quantisation volume (i.e. all of space)
2) When we wish to consider one single direction, so cannot integrate over solid angles

Does anyone here know how to deal with these situations, or of an alternative, more useful, definition?

I have tried solving 1) by choosing an appropriate quantisation volume, but the answer one gets is rather dependent upon the volume chosen.

I have seen a book solve 2) by forming an intensity of emitted radiation instead of a rate, but this is of no use for my purposes (an intensity includes units $$m^{-2}$$, making it impossible to recover a rate in $$s^{-1}$$ because a single direction has zero area).
I have tried to solve 2) by invoking uncertainty principles, but this is a bit hand waving and not very rigorous.


text:
## Wednesday, 28 February 2018

### On A Mathsy St David's Day

Q: What links Bertrand Russell, the equals sign, the word in the English language with the most Zs, the use of π as the ratio of a circle's circumference to its diameter, the Online Encyclopedia of Integer Sequences, and the number 36?

A: Wales.

By way of explanation, and in mathsy celebration of St David's Day, I offer below a handful of mathematical somethings with a Welsh bent, peppered with one or two suggested explorations or diversions that teachers may wish to share and use with students on the day or, indeed, any other.

Dydd Gŵyl Dewi Hapus.

The Equals Sign

Welsh Doctor and mathematician Robert Recorde, born in 1510 in Tenby, Pembrokeshire, South Wales, was a popular author of a number of mathematical books — which he wrote, unusually for the time, in the English vernacular, thus making his writing more accessible than most scholarly books of the age, which were usually written in Latin.

In his 1557 book, The Whetstone of Witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Nombers [1] (which you can peruse electronically here), Recorde 'invented' the '=' symbol, 'to avoide the tediouse repetition of these woordes : is equalle to :', which he had already used some 200 times in the book [2].

During Recorde's time, much of the mathematical notation we take for granted today was not yet in use.  In designing the symbol '=' — 'a paire of paralleles, or Gemowe [twin] lines of one length, thus: =====, bicause noe 2 thynges can be moare equalle' — Recorde's initial motivation to abbreviate was quickly overtaken by something more profound, more enduring.  As Joseph Mazur (2014) eloquently puts it, 'the concise character of the symbol came with an unintended benefit: it enabled an unadorned picture in the brain that could facilitate comprehension'.

Recorde was also, for example, and in the same book, the first to use the plus and minus signs in English: 'There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse'.  And neither was there an easy way in the 16th century of denoting the powers of numbers, so Recorde coined the now unsurprisingly obsolete term ‘zenzizenzizenzic' [3] to ‘doeth represent the square of squares squaredly’, or in other words to denote the square of the square of a number's square:

${\left( {{{\left( {{n^2}} \right)}^2}} \right)^2} = {n^8}$
Recorde also used another word (which didn't quite catch on), the 'sursolid', meaning to be raised to a prime number greater than three.  So a power of five would be the first sursolid, a power of seven the second sursolid, a power of eleven the third, and so on.

Suggested explorations/diversions with/for students:
• Find the zenzizenzizenzic of n for 0 < n < 10.
• Devise questions in Recordian notation and answer them, for example: 'What is the fourth sursolid of two divided by the zenzizenzizenic of two?  Give your answer in modern and Recordian form'.
• Consider and explore the difference between:

${\left( {{{\left( {{n^2}} \right)}^2}} \right)^2}\;{\rm{and}}\;{n^{{2^{{2^2}}}}}$

The first use of π to denote C/d

Before being denoted π, the ratio of the circumference of a circle to its diameter was referred to typically in the Latin 'quantitas in quam cum multiflicetur diameter, proveniet circumferencia ('the quantity which, when the diameter is multiplied by it, yields the circumference)' (Rothman, 2009).

Welsh mathematician William Jones, born in 1675 in Llanfihangel Tre'r Beirdd, on the Isle of Anglesey, North Wales, was the first person to use π to denote the ratio of a circle's circumference to its diameter, doing so in his 1706 book Synopsis palmariorum matheseos: or, A new introduction to mathematics: containing the principles of arithmetic & geometry demonstrated, in a short and easie method; with their application to the most useful parts thereof ... Design'd for the benefit, and adapted to the capacities of beginners [4].  You can peruse the book electronically here [5].

This first ever appearance of π denoting the ratio of a circle's circumference to its diameter can be seen on p243, then more explicitly on p263, as excerpted below.  It can also be seen that Jones gave π correct to 100 decimal places, 'as Computed by the Accurate and Ready Pen of the Truly Ingenious Mr. John Machin', using an infinite series whose sum converged to π (see this on 'Machin's Formula' from Peter Rowlett in The Aperiodical).

Using π in this way was a significant philosophical step forwards; Jones was more than merely abbreviating.  Although unable to prove it, Jones recognised that the ratio of a circle's circumference to its diameter could not be expressed as a rational number — or in other words, that π was an irrational number — as can be seen in the p243 excerpt above: 'For as the exact Proportion between the Diameter and the Circumference can never be expres'sd in Numbers' [6].  Jones recognised, as such, that 'to represent an ideal that can be approached but never reached.... only a pure platonic symbol would suffice' (Rothman, 2009).

Jones' use of π as C/d was popularised when the great Swiss mathematician Leonhard Euler adopted it in his Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) in 1748 [7].

Suggested explorations/diversions with/for students:
• Share the excerpts above with students and try to make sense of them together.
• Have students work out their π-related birthdays, past or future, discussing precision (see this post).
• Note, for example, that at some point on St Davids' Day 2018:
• 3 year-olds born on 8 January 2015, will be π years old.
• 9 year-olds born on 17 April 2008, will be ππ years old.
• 36 year-olds born on 8 January 2015, will be ππ years old.

The OEIS

Mathematician Neil Sloane, born on 10 October 1939 in Beaumaris, North Wales — described by Erica Klarreich as 'The Connoisseur of Number Sequences' in this article in Quanta magazine, and as 'the Guy who Sorts All the World's Numbers in his Attic' in this reprint of the article in Wired — is considered by some to be one of the most influential mathematicians of our time, because in 1964 Sloane founded The Online Encyclopedia of Integer Sequences (OEIS).

The OEIS, as the name suggests — or Sloane, as it is often referred to by its users — is an online database of at the current count, over one quarter of a million integer sequences.  It is designed to be used by researchers in mathematics, but as John Conway and Tim Hsu put it in 2006, 'most Nerds should be able to get some enjoyment out of it'.

Enjoy this short selection of some gems that I first discovered through the OEIS:

And consider this sequence (sequence A168087): a(n) = the smallest number whose Welsh name (masculine or feminine versions) in the modern Decimal System contains n letters of the alphabet.  (For example, 224, dau gant dau ddeg pedwar, is the 20th number in the sequence, and is the smallest number with 20 letters.)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 7 4 11 12 15 16 14 27 24 47 44 127 124 147 144 244

Suggested explorations/diversions with/for students:
• Try and ascertain the rule that describes each of my selection of 'gems' above.
• For sequence A168087
• Continue the sequence to n = 40.
• Find a(100).
• Do the same for sequence A168085 (using the traditional Vigesimal System).
• Generate the same sequence for numbers in English, and other languages.
• Describe this sequence 4, 2, 3, 3, 6, 4, 6, 5, 4, 3, 3, 8, 5... (sequence A140396), entered into the OEIS by Sloane himself in 2008, perhaps as a nod to his Welsh heritage.
• Generate a sequence of numbers that have the same amount of letters in Welsh as in English.  For example, a(1) = 2, because 'two' in English and 'dau' in Welsh has 3 letters.
• Maybe submit the sequence to the OEIS (it's not there; I've checked) on behalf of a student (with parental consent of course) who generates it and defines it best, according to the OEIS' format.
• Try this puzzle set by Sloane in Quanta Magazine:
• Can you figure out the 'simple' rule that describes this sequence 13, 26, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, … (click here for the solution when you're ready).
• The image below shows the 'zigzag triangle', via JohnConway and Tim Hsu (2006).  On the LHS of the triangle are the Zig (or secant or Euler) numbers, and on the RHS are the Zag (or tangent) numbers.
• Find the next Zig and Zag numbers.
• How far can you keep going?

T

The Number 36

Charles Yang, associate professor in the University of Pennsylvania Department of Linguistics, and not Welsh, has shown (2005, cf. 2015) that if there is a linguistic rule, a generalisation in other words that can be applied to a set of N words, but within this set of N words there is a subset of words, e, that do not follow this rule and that must therefore be memorised, then

$e < {\theta _N}\;where\;{\theta _N}: = \frac{N}{{\ln \left( N \right)}}$
This model, dubbed the 'Tolerance Principle', can be applied to how we, as children, learn to count.  In short (see this post, 'On 73'for more detail), by using the Tolerance Principle, we can find the least amount of words that we need to learn ‘to overcome the exceptions we have to memorise’.  In Welsh, in the modern Decimal System of counting, the numbers 1 to 10 are the only 'exceptions': Un (one), Dau (two), Tri (three), Pedwar (four), Pump (five), Chwech (six), Saith (seven), Wyth (eight), Naw (nine), and Deg (ten).  All numbers beyond this are generalised from them, for example eleven is un deg un (one ten one), twelve is un deg dau (one ten two), seventy three is saith deg tri (seven ten three), etc.  Thus, the smallest value of N in Welsh such that θN = 10 is 36:

\begin{array}{l}\begin{align}N &= 36\;\\\because10 &= e < {\theta _N} = \frac{{36}}{{\ln \left( {36} \right)}}\\\;where\;{\rm{ }}\frac{{36}}{{\ln \left( {36} \right)}} &= 10.045991...\end{align}\end{array}
Or in other words, once a child has learned to count to 36 in Welsh, they have learned the rules of the game sufficiently to overcome the cognitive need for memorisation, and thus to keep going.

Bertrand Russell

Bertrand Russell, the mathematician, logician and humanist, was born on 18 May 1872, in Trellech, Monmouthshire.  Russell lived for most of his later years at Plas Penrhyn in Penrhyndeudraeth, Merioneth, North Wales, with a view south to Cardigan Bay and north to the mountains of Eryri (Snowdownia).  He died at Plas Penrhyn on February 2, 1970 (read his obituary in the New York Times here), was cremated at Colwyn Bay and had his ashes scattered over the Welsh hills.

Russell won a scholarship to read mathematics at Trinity College, Cambridge University, and with Alfred North Whitehead wrote his monumental three-volume work, Principia Mathematica, between 1910 and 1913.  ('Logicomix: An Epic Search for Truth', a wondrous graphic novel 'inspired by the epic story of the quest for the Foundations of Mathematics', described Russell and Whitehead's Principia as 'a heroic intellectual adventure.')   In 1950, Russell was made a Nobel Laureate in Literature “in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought”.

Read a short biography of Russell here, and watch his 1959 BBC 'Face to Face' interview with John Freeman here, and/or read the transcript here.

[1] Ian Stewart (2009, p20) explains that 'Cossike practise; refers to algebra: the Reanaisssance Italian algebraists refered to the unknown, which we now call x, as cosa, Italian for 'thing'.  As in cosa nostra, 'this thing of ours', referring to the Mafia.  'Surde nombers' are things like square roots, and the word 'surd' still exists in English, though it is seldom used nowadays'.

[2] For a little more, see this from Mark Dominus and this from Samuel Arbesman in Wired.

[3] Zenzizenzizenzic is also has more Zs than any other word in the English language (According to the Oxford English Dictionary).

[4] Phew!

[5] Jones was also noted for his 1711 work, with assistance from Newton, Analysis per quantitatum series, fluxiones ac differentias, available to view digitally here, in which he introduced the dot notation for differentiation in calculus.

[6] π was proven to be irrational by Johann Heinrich Lambert in 1761.  See this from The World of Pi website.

[7] For more on William Jones and the origins of π, in addition to Rothman's excellent 2009 History Today article, see this as equally excellent Guardian article from Gareth Ffowc Roberts.

Arbesman, S.  (2014, 30 October).  'The Invention of the Equals Sign'.  Wired.  Retrieved February 2018 from https://www.wired.com/2014/10/invention-equals-sign/

BBC Wales Arts.  (2008, 15 December).  Bertrand Russell.  Retrieved February 2018 from http://www.bbc.co.uk/wales/arts/sites/bertrand-russell/

BHA Website.  (n.d.).  Plas Penrhyn, Wales.  In Humanist Heritage.  Retrieved February 2018 from http://humanistheritage.org.uk/articles/plas-penrhyn-wales/

Cavanaugh, R.  (2016, March 4).  'Robert Recorde, Inventor of the equals sign', Mental Floss.  Retrieved February 2018 from http://mentalfloss.com/article/76342/retrobituaries-robert-recorde-inventor-equals-sign

Conway, J.H., and Hsu, T.  (2006).  Some Very Interesting Sequences.  Retrieved January 2018 from http://www.math.sjsu.edu/~hsu/pdf/sequences.pdf
Dominus, M.  (2006, April 7).  Robert Recorde Invents the Equals Sign [Blog post].  Retrieved February 2018 from https://blog.plover.com/math/recorde.html

Doxiadis, A. and Papadimitriou, C.  (2009).  Logicomix: An Epic Search for Truth.  London: Bloomsbury.

Ffowc Roberts, G.  (2015).  'Pi Day 2015: meet the man who invented π', The Guardian.  Retrieved February 2018 from https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/14/pi-day-2015-william-jones-the-welshman-who-invented-pi

Gourévitch, B.  (2013, 13 April).  Johann Heinrich Lambert (1728 - 1777) [Blog post].  Retrieved February 2018 from http://www.pi314.net/eng/lambert.php

Irvine, A.D.  (1996, December).  'Bertrand Arthur William Russell', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Russell.html

Jones, W.  (1706).  Synopsis Palmariorum MatheseosOr, A New Introduction to the Mathematics: Containing the Principles of Arithmetic and Geometry.  London.  Retrieved February 2018 from https://ia801003.us.archive.org/9/items/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/Synopsis_Palmariorum_Matheseos.pdf

Johann Heinrich Lambert.  (n.d.).  In Wikipedia.  Retrieved February 2018 from https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert

Jones, W.  (1711).  Analysis per quantitatum series, fluxiones ac differentias.  London.  Retrieved February 2018 from https://archive.org/details/analysisperquan00jonegoog

Klarreich, E.  (2015).  'The Connoisseur of Number Sequences', Quanta Magazine.  Retrieved February 2018 from https://www.quantamagazine.org/neil-sloane-connoisseur-of-number-sequences-20150806/

Mazur, J.  (2014).  Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers.  Princeton: Princeton University Press.

O'Connor, J.J. and Robertson, E.F.  (2003, April).  'John Machin', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Machin.html

O'Connor, J.J. and Robertson, E.F.  (1998, September).  'Leonhard Euler', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Euler.html

O'Connor, J.J. and Robertson, E.F.  (2002, April).  'Robert Recorde', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Recorde.html

O'Connor, J.J. and Robertson, E.F.  (2005, February).  'William Jones', School of Mathematics and Statistics, University of St Andrews.  Retrieved February 2018 from http://www-history.mcs.st-and.ac.uk/Biographies/Jones.html

Plus and minus signs.  (n.d.).  In Wikipedia.  Retrieved February 2018 from https://en.wikipedia.org/wiki/Plus_and_minus_signs

ReasonPublic (2012, March 13).  Bertrand Russell - Face to Face Interview (BBC, 1959) [Video File].  Retrieved February 2018 from https://youtu.be/1bZv3pSaLtY

Recorde, R.  (1557).  The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Cossike practise, with the rule of Equation: and the woorkes of Surde Nombers.  London.  Retrieved February 2018 from https://archive.org/details/TheWhetstoneOfWitte

Rothman, P.  (2009).  'William Jones and his Circle: The Man who Invented Pi', History Today, Vol. 59, Issue 7.  Retrieved February 2018 from https://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi

Rowlett, P.  (2015, March 13).  'Pi Approximation: Machin's Formula'.  The Aperiodical.  Retrieved February 2018 from http://aperiodical.com/2015/03/pi-approximation-machins-formula/

Springer, M.  (2012, 31 August).  Face to Face with Bertrand Russell: ‘Love is Wise, Hatred is Foolish’.  In Open Culture.  Retrieved February 2018 from http://www.openculture.com/2012/08/face_to_face_with_bertrand_russell_love_is_wise_hatred_is_foolish.html

Stewart, I.  (2009).  Professor Stewart's Hoard of Mathematical Treasures.  London: Profile Books.

The Whetstone of Witte.  (n.d.).  In Wikipedia.  Retrieved February 2018 from https://en.wikipedia.org/wiki/The_Whetstone_of_Witte

Whitman, A.  (1970, 3 February).  Bertrand Russell is Dead; British Philosopher, 97.  Retireved February 2018 from http://www.nytimes.com/books/01/04/29/specials/russell-obit.html

Yang, C.  (2015).  'Negative Knowledge from Positive Evidence', Language, Vol. 91, No.4, pp938-953.  Retrieved from: https://www.linguisticsociety.org/sites/default/files/07_91.4Yang.pdf

Yang, C.  (2005).  'On Productivity', in Linguistic Variation Yearbook 5, John Benjamins Publishing Company.  Retrieved from: http://www.ling.upenn.edu/~ycharles/papers/l.pdf

Zenzizenzizenzic.  (n.d.).  In Wikipedia.  Retrieved February 2018 from https://en.m.wikipedia.org/wiki/Zenzizenzizenzic


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# Forum: FPGA, VHDL & Verilog Viterbi Decoder

Rate this post
 0 ▲ useful ▼ not useful
I'm sorry if I have broken the rules by making this a general post. I am
attempting to create a decoder for convolution-encoded parity arriving,
and needing to be processed, at a rate of 100Mbits/s, this being my
preferred clock frequency for the Zynq 7045 SOC I am using for the job.
I have elected to make my first attempt a register exchange engine,
bearing in mind that some academics have proposed methods by which the
physical exchange of registers might be avoided.

My question, however is related to whether this device should operate on
blocks, or in a continuous mode, in case there is any statistical merit
in operating the decoder in a manner in which it outputs a bit every
clock, based on a path metric obtained by subtracting the branch metrics
associated with the beginning of a path, and adding those calculated
most recently. The comparison of path metrics for each state being
performed every clock cycle. Such a comparison, in the case of my
application, being of 64 words, probably requiring some degree of
pipelining, necessary information needing to be stored for the depth
thereof.

Given that it might be determined that L state transitions are needed to
compute a reliable estimate of the last L uncoded information bits, and
that the method I am speculating would discard all but the earliest of
these, I have doubts regarding its validity, a problem, for me, being
that, with a moving trellis of finite length L, initial conditions are
inherited, rather than assumed, it seeming to me that they should be
reset at eack clock tick, and costs recalculated over all states
(columns) in the trellis. The latter making the implementation demanding
of resources probably unavailable. My decision length, L, is 128, and
there are, as mentioned, 64 states.

The large decision length is determined by the considerable puncturing
of the 4-bit mother parity code required by the customer. Code rates
from 8/9 to 1/4 need to be supported.

Any help would be appreciated.
Best regards
Julian

• $formula (LaTeX syntax)$


text:
# Thread: Related Rates problems

1. ## Related Rates problems

I am greatly struggling with related rates problems. I'll give an example and perhaps someone can walk me through it.

At noon, ship A is 70 km west of ship B. Ship A is sailing south at 25 km/h and ship B is sailing north at 15 km/h. How fast is the distance between the ships changing at 6:00 PM?

I don't even know where to begin really, according to my notes I should begin by writing down the given info and what I'm trying to find and drawing a picture of applicable, but after doing all this I still have no idea how to go about finding the answer. I know that da/dt would be 25 and db/dt would be 15, but I don't know how I would use that information at all. I also know that a triangle will be involved, but again I don't know how to use that information to solve this problem.

2. Ok so bottom line, based on the diagram, you want to find $\frac{dz}{dt}$ when $y=240$ - that's your question

So at 6pm, 6 hrs have passed; therefore, ship A has travelled 150km south (y1) and ship B has travelled 90 km north (y2). y= y1+y2=240km

x = 70km = CONSTANT

AT THE INSTANT of 6pm, z would be $\sqrt{240^2+70^2}=250$ km

Therefore, to solve this question, you use a modified version of pythagorean's theorem: $z^2=x^2+(y_1+y_2)^2$
Note: $y_1$ and $y_2$ are variables, NOT constants.

$z^2=x^2+(y_1+y_2)^2 \to 2z\frac{dz}{dt}=2x\frac{dx}{dt}+2(y_1+y_2)\left(\f rac{dy_1}{dt}+\frac{dy_2}{dt}\right)$

Plug in all your numbers and solve... should get $\frac{192}{5}$ km/h

3. I'm still very very confused. I understood everything up to the final two lines, the equation involving z^2=etc. I have no iea how you came up with that to solve it? I understand that the distance between the ships at 6pm is 250 km but I don't understand how to use that information to find the rate of change between the ships at that time.

4. Originally Posted by fattydq
I'm still very very confused. I understood everything up to the final two lines, the equation involving z^2=etc. I have no iea how you came up with that to solve it? I understand that the distance between the ships at 6pm is 250 km but I don't understand how to use that information to find the rate of change between the ships at that time.
you need to find a relationship between z, x, and y...there's a right angle triangle... $z^2=y^2+x^2$ seem fimilar to you?

5. Yes of course, but he already used the pythagorean theorem to find the 250km, I don't see how using the pythagorean theorem could possibly lead to the answer of the fraction he provided

6. Originally Posted by fattydq
I am greatly struggling with related rates problems. I'll give an example and perhaps someone can walk me through it.

At noon, ship A is 70 km west of ship B. Ship A is sailing south at 25 km/h and ship B is sailing north at 15 km/h. How fast is the distance between the ships changing at 6:00 PM?
let ship A begin at the origin. its position as a function of time is ...

$x = 0$

$y = -25t$

ship B starts at (70,0). its position as a function of time is ...

$x = 70$

$y = 15t$

using the distance formula, the distance between the two ships at any time t is ...

$D = \sqrt{(70-0)^2 + [15t - (-25t)]^2}$

$D = \sqrt{4900 + 1600t^2}$

$\frac{dD}{dt} = \frac{1600t}{\sqrt{4900+1600t^2}}$

now ... evaluate $\frac{dD}{dt}$ for $t = 6$ hrs

7. Originally Posted by skeeter
let ship A begin at the origin. its position as a function of time is ...

$x = 0$

$y = -25t$

ship B starts at (70,0). its position as a function of time is ...

$x = 70$

$y = 15t$

using the distance formula, the distance between the two ships at any time t is ...

$D = \sqrt{(70-0)^2 + [15t - (-25t)]^2}$

$D = \sqrt{4900 + 1600t^2}$

$\frac{dD}{dt} = \frac{1600t}{\sqrt{4900+1600t^2}}$

now ... evaluate $\frac{dD}{dt}$ for $t = 6$ hrs
Again I understand the whole thing until you involve the final step, the fraction.

8. take the derivative of D w/r to t using the chain rule ...

$D = (4900 + 1600t^2)^{\frac{1}{2}}$

$\frac{dD}{dt} = \frac{1}{2}(4900 + 1600t^2)^{-\frac{1}{2}} \cdot 3200t$

$\frac{dD}{dt} = 1600t(4900 + 1600t^2)^{-\frac{1}{2}}$

$\frac{dD}{dt} = \frac{1600t}{\sqrt{4900 + 1600t^2}}$

clear enough?


text:
# The sum of all natural numbers: 1 + 2 + 3 + 4 + .... Part 2

Many people know that

But in reality

Let us consider in more detail the first result. Of course, a series of natural numbers diverges in the classical sense (in the sense of convergence of a sequence of partial sums: it, of course, has no limit). In this article, the author mentions other summation methods, such as the Cesaro method and the Abel method. Here are some examples: the sum of such a series

using the cesaro method will be equal $\ dfrac12$.

Another example:

In my opinion, it is wrong to say that the sum of the first row is equal to $\ dfrac12$; correctly say that the sum of the first row in the sense of Cesaro is equal to $\ dfrac12$. Similarly for the second: its sum in the sense of Abel is equal to $\ dfrac14$.

In view of this, in the first result (that $- \ dfrac {1} {12}$) there is a substitution of concepts, which leads to a contradiction with common sense.

We now consider in more detail the second result. First, we denote the entire amount for $X$$X$:

Now we perform the following transformations:

From here

There is another solution. Combine the terms in another way:

i.e

In fact, starting from the top three, we can distinguish 7 terms, the sum of which will be 49, and we will come to the equation

$1+2+3+49X=X,$

which will give the same result.

In general, you need to act like this: select the first $n$$n$terms, and then in parentheses take $2n+1$$2n + 1$terms:

Arithmetic progression $1 + \ dots + n$is equal to $\ dfrac {n (n + 1)} {2}$, therefore, we obtain the equation

where does it turn out that

All Articles


text:
# Combination of proposals¶

Given the two sets of bounding box obtained through the processing of depth and intensity video streams, we compute the following pairwise overlap ratios:

$$r = \frac{area \left(B_{curr} \cap B_{acc} \right)}{area \left(B_{curr} \cup B_{acc} \right)}$$

A possible luggage, obtained through the formula above, is no longer detected because of two possible reasons:

• a left item is removed from the scene
• the item detected is standing still for a long amount of time. After this time the item became part of the depth and rgb background. When the item became part of the background model we can’t detect its presence doing $$current\_frame - bg\_model$$ so we need a way to retain the information previously discovered. If a Bounding box is present at the frame t-1 but not in the frame t, we check if pixels in the area, defined by his bounding box, are still the same (i.e. the luggage is still there): this check is performed by using the normalized correlation between the pixel in the t-1 and t frames. If the similarity is above a certain threshold (i.e. 0.9) we keep drawing the old bounding box.


text:
MathSciNet bibliographic data MR1860064 (2003a:46095) 46L55 (37D20 46L05) Hou, Chengjun; Chen, Xiaoman A note on the ideals of groupoid \$C\sp *\$$C\sp *$-algebras from Smale spaces. Bull. Austral. Math. Soc. 64 (2001), no. 2, 271–279. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# FHSST Physics/Electrostatics/Charge

Electrostatics The Free High School Science Texts: A Textbook for High School Students Studying Physics Main Page - << Previous Chapter (Heat and Properties of Matter) - Next Chapter (Electricity) >> Definition - Charge - Electrostatic Force - Electric Fields - Electrical Potential - Important Equations and Quantities

# Charge

All objects surrounding us (including people!) contain large amounts of electric charge. Charge can be negative or positive and is measured in units called coulombs (C). Usually, objects contain the same amount of positive and negative charge so its effect is not noticeable and the object is called electrically neutral. However, if a small imbalance is created (i.e. there is a little bit more of one type of charge than the other on the object) then the object is said to be electrically charged.

Some rather amusing examples of what happens when a person becomes charged are for example when you charge your hair by combing it with a plastic comb and it stands right up on end! Another example is when you walk fast over a nylon carpet and then touch a metal doorknob and give yourself a small shock (alternatively you can touch your friend and shock them!)

Charge has 3 further important properties:

• Charge is always conserved. Charge, just like energy, cannot be created or destroyed.
• Charge comes in discrete packets. The smallest unit of charge is that carried by one electron called the elementary charge, e, and by convention, it has a negative sign ( ${\displaystyle e=-1.6\times 10^{-19}{\rm {C}}}$ ).
• Charged objects exert electrostatic forces on each other. Like charges repel and unlike charges attract each other.

 The word `electron' comes from the Greek word for amber! The ancient Greeks observed that if you rubbed a piece of amber, you could use it to pick up bits of straw. (Attractive electrostatic force!)

You can easily test the fact that like charges repel and unlike charges attract by doing a very simple experiment. Take a glass rod and rub it with a piece of silk, then hang it from its middle with a piece string so that it is free to move. If you then bring another glass rod which you have also charged in the same way next to it, you will see the rod on the string turn away from the rod in your hand i.e. it is repelled. If, however, you take a plastic rod, rub it with a piece of fur and then bring it close to the rod on the string, you will see the rod on the string turn towards the rod in your hand i.e. it is attracted!

What actually happens is that when you rub the glass with silk, tiny amounts of negative charge are transferred from the glass onto the silk, which causes the glass to have less negative charge than positive charge, making it positively charged. When you rub the plastic rod with the fur, you transfer tiny amounts of negative charge onto the rod and so it has more negative charge than positive charge on it, making it negatively charged.

## Conductors and Insulators

Some materials allow charge carriers to move relatively freely through them (e.g. most metals, tap water, the human body) and these materials are called conductors. Other materials, which do not allow the charge carriers to move through them (e.g. plastic, glass), are called non-conductors or insulators.

Aside: As mentioned above, the basic unit of charge, namely the elementary charge, e, is carried by the electron. In a conducting material (e.g. copper), when the atoms bond to form the material, some of the outermost, loosely bound electrons become detached from the individual atoms and so become free to move around. The charge carried by these electrons can move around in the material! In insulators, there are very few, if any, free electrons and so the charge cannot move around in the material.

If an excess of charge is put onto an insulator, it will stay where it is put and there will be a concentration of charge in that area on the object. However, if an excess of charge is put onto a conductor, the charges of like sign will repel each other and spread out over the surface of the object. When two conductors are made to touch, the total charge on them is shared between the two. If the two conductors are identical, then each conductor will be left with half of the total charge.


text:
## Corrigendum to: Conservativity of time-flow invertible and boundary control systems.

On p. 28 l. -10: Say "bounded open connected set (bounded domain)" instead of "open bounded set". This assumption is required for the Poincare's inequality to hold. It is clear that this assuption of connectness can be relaxed; The point is not to allow $\Omega$ have a component $\Omega'$, such that $\partial \Omega' \subset \Gamma_1$.

On p. 29 l. -11: Remove the sentence starting "In the same manner...". Its proof is wrong, the statement is untrue, and it is never used anywhere.

There is letter $h$ in many formulas on p. 29 -- 30. It should be $z_0$.

There is a number of typos etc. all over the paper that I will not try to correct here.


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# vectors

• August 31st 2006, 10:55 AM
ligekron
vectors

collinearity

proving collinearity

for the high school level
• August 31st 2006, 12:08 PM
OReilly
Quote:

Originally Posted by ligekron

collinearity

proving collinearity

for the high school level

Collinearity is refering to position of points in line.

Given points are collinear if line contains all of them. If line doesn't contain all of them then they are non-collinear.
• August 31st 2006, 01:05 PM
Soroban
Hello, ligekron!

Quote:

Please explain: (1) Collinearity, (2) Proving collinearity
for the high school level

Since your heading says "Vectors", I assume a vector approach is in order.

Three points $A,\,B,\,C$ are collinear (lie on a straight line)
if any two of their vectors $\{\overrightarrow{AB},\;\overrightarrow{BC},\; \overrightarrow{AC}\}$ are scalar multiples of each other.

$\text{Example: }\;A(2,3),\;B(3,5),\;C(5,9)$

$\text{Let }\vec{u} \:=\:\overrightarrow{AB} \:= \:\langle 3,5\rangle - \langle 2,3\rangle \:=\:\langle 1,2\rangle$

$\text{Let }\vec{v}\:=\:\overrightarrow{BC} \:=\:\langle 5,9\rangle - \langle 3,5\rangle \:=\:\langle 2,4\rangle$

$\text{Since }\vec{v} = 2\vec{u}\text{, points }A,\;B,\,C\text{ are collinear.}$


text: i The use of superhydrophobic surfaces is useful but can be further improved by using new lubricant-impregnated surfaces. Hydrophobic Mesh air filters offer a water-repellant, low pressure drop media to reduce ingress of airborne mist and other liquids on indoor and outdoor enclosures deployed in harsh environments. Ultra stable Current solutions RayOne Hydrophobic Current solutions Patient outcomes • eerng f eni - tsGli material • Stable positioning • Aspheric design However, a hydrophilic surface (one that has an original contact angle less than 90°) becomes more hydrophilic when microstructured – its new contact angle becomes less than the original. Cyclopentasiloxane is the component of antiperspirant that functions as the hydrophobic element to this project. Ultrahydrophobic (or superhydrophobic) surfaces are highly hydrophobic, i.e., extremely difficult to wet. θ = The slide angle is another dynamic measure of hydrophobicity and is measured by depositing a droplet on a surface and tilting the surface until the droplet begins to slide. An example of superhydrophobic effect in live application is the team Alinghi in America's Cup using specially treated sailing jackets. 0 The enhanced performances are mainly owing to the activation of the microscale edge effect in the hierarchical surface, which increases the energy barrier for ice bridging as well as engendering the liquid lubrication during the deicing/defrosting process. These micro-flowers offer nanoscale features which enhance the surface's hydrophobicity, without the use of low surface energy coatings. where φ is the area fraction of the solid that touches the liquid. [29][30][31][32][33], Research in superhydrophobicity recently accelerated with a letter that reported man-made superhydrophobic samples produced by allowing alkylketene dimer (AKD) to solidify into a nanostructured fractal surface. more spherical shape of the droplets. 0 A special family of nano-powders was developed, capable of forming surfaces with exceptional super-hydrophobic (or ultra-hydrophobic) properties. r ) {\displaystyle \theta _{a}=\lambda _{p}(\theta _{a,0}+w)+(1-\lambda _{p})\theta _{air}}, θ Some birds are great swimmers, due to their hydrophobic feather coating. + ) ( Ultrahydrophobic (or superhydrophobic) surfaces are highly hydrophobic, i.e., extremely difficult to wet. Since the product is hydrophobic, Ultra-Ever Dry also eliminates ice formation. λ The shape of the water droplets and either hydrophobic or hydrophilic properties of a surface are determined by the "contact angle". a θ ( A simple model can be used to predict the effectiveness of a synthetic micro- or nano-fabricated surface for its conditional state (Wenzel or Cassie-Baxter), contact angle and contact angle hysteresis. r UV light creates electron-hole pairs, with the holes reacting with lattice oxygen creating surface oxygen vacancies while the electrons reduce V5+ to V3+. Contact angle hysteresis is a phenomenon that characterizes surface heterogeneity. media caption Super-hydrophobic material The faster raindrops bounce off an anorak, the drier it stays. θ + Ultra Hydrophobic Coating Turns Cars Into Anti-Scratch, Anti-UV Light, Self-Cleaning $18.50 –$ 45.00; Super Hydrophobic Coating Protects Wood Window From Water Mold Mildew $12.50 –$ 28.00; Protect Ceramic Sinks from Scratches and Stains, Easy-Clean Sink - Superhydrophobic Nano Coating $12.50 –$ … [65], The ability of packaging to fully empty a viscous liquid is somewhat dependent on the surface energy of the inner walls of the container. Surfaces prepared offer contact angles higher than 160° with typical sliding angles around 10°. {\displaystyle \Lambda _{C}={\frac {-\rho {g}{V^{1/3}}{\Big (}{\Big (}{\frac {1-cos(\theta _{a})}{sin(\theta _{a})}}{\Big )}{\Big (}3+{\Big (}{\frac {1-cos(\theta _{a})}{sin(\theta _{a})}}{\Big )}^{2}{\Big )}{\Big )}^{2/3}}{(36\pi )^{1/3}\gamma cos(\theta _{a,0}+w-90)}}}. θ To understand the behavior of a surface towards water, its contact angle (CA) has to be measured, which will provide information on the interaction energy between the surface and the liquid. p Contact angle is a measure of static hydrophobicity, and contact angle hysteresis and slide angle are dynamic measures. In some cases, Gentoo can be removed using abrasive materials such as heavy-duty Scotch Brite™ (which contains aluminum oxide) or sandpaper. Dettre and Johnson discovered in 1964 that the superhydrophobic lotus effect phenomenon was related to rough hydrophobic surfaces, and they developed a theoretical model based on experiments with glass beads coated with paraffin or TFE telomer. How does Ultra-Ever Dry® work? Because the natural world is full of hydrophobic and hydrophilic surfaces, the basics of the phenomenon have been known by scientists for at least two centuries. a ) Researchers have used lasers to create an extremely hydrophobic material with potential applications from sanitation to solar panels. [39] Debates have recently emerged concerning the applicability of the Wenzel and Cassie-Baxter models. 0 It gives the leaves unique sefl-cleaning properties, as the droplets pick up dirt particles while rolling away. [1] This is also referred to as the lotus effect, after the superhydrophobic leaves of the lotus plant. − By extended storage in the dark, water is replaced by oxygen and hydrophilicity is once again lost. 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