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# Measuring Domestic Output and National Income

The countries we find ourselves living in are so insanely massive; how can one place have hundreds of thousands of businesses and people all engaging in various economic activities? Who keeps track of all their output, and what does it mean for the country as a whole? In this explanation, we'll go into detail about measuring domestic output and national income. If you are ready to join us on this epic quest, keep on reading!

## Types of National Income

National income refers to the total accounting that occurs in an economy, and knowing the many types of national income is a part of figuring it out.

National income is a measure of the overall performance of an economy.

The organization that gathers data for the national income accounts is the Bureau of Economic Analysis, an agency of the commerce department. They developed a report called the National Income and Products Accounts (NIPA). Information gathered in these reports is used to:

• Analyze the health of the economy at various production levels

• Track economic growth or decline

• Plan policies to ensure the long-term economic health

The national income can be used to calculate the gross domestic product (GDP). This is called the income approach.

The income approach takes into account many sources of income, such as compensation of employees, rents, interest, proprietors' income, corporate profits, and taxes on production and imports. All of these income sources combined make up a country's national income, often referred to as the gross national income (GNI).

The gross national income for the United States in 2021 was $23,374,710,000,000 or 23.3 trillion dollars. The 2021 GNI was significant because it was a 9.94% increase from the previous year. Can you see your personal income reflected as a share of the national income? 1 For more information on national income and other related topics, check out our explanation of National Accounts! ## Measures of National Income and Output Let's discuss measures of national income and output! National income is the summation of individual, business, and government cash inflows. It is a way of measuring output by determining how much money the output was purchased for, at which point it becomes the seller's income. Using data from FRED (Federal Reserve economic data), the following breakdown of the real national income is provided.  Total Real National Income$19,937.975 billion Compensation of employees $12,598.667 billion Proprietor's income$1,821.890 billion Rental income $726.427 billion Corporate profits$2,805.796 billion Net interest and miscellaneous $686.061 billion Taxes on production and imports$1,641.138 billion

Table 1 - Federal Reserve economic data real national income by category2

Using this data (and a few smaller categories left out), one can calculate GDP with the income approach.

The income approach formula uses the value of a nation's income, which includes wages, profits, interest, and rent.

The formula for the income approach is as follows:

$$\hbox{GDP} = \hbox{Total Wages + Total Profits +Total Interest + Total Rent + Proprietors income + Taxes}$$

## Measurement of National Output and Income

Countries and those who are accountable for the ongoing economic success closely monitor the measurement of national output and income. They are strong indicators of the total productive capacity of a country and how it changes year to year. There are a few ways of calculating it, whether it's the final cost of the good, the value added at each step, or the total incomes of individuals and businesses.

No matter which approach is used to calculate GDP, they all provide a total valuation of the productive power of an economy. Having a strong GDP and GDP growth will entice other countries to invest and engage in business transactions.

For an in-depth explanation of measuring GDP, consider reading our explanation of Measured GDP

## Domestic Output Formula

The domestic output formula refers to the GDP or gross domestic product of a nation. The three main ways of calculating GDP are the income approach, the expenditures approach, and the value-added approach.

The income approach calculates the total productive output by summing total income across an economy. The income approach totals the inflows of cash from providing goods or services. Similar to calculating the total cost of output, this approach comes from the opposite direction, calculating the money paid for output that becomes income. This approach has the added benefit of automatically including foreign purchases of domestic goods without imports needing to be calculated.

The income approach totals incomes across several categories; employees' wages, proprietors' income, corporate profits, rent, interest, and taxes on production and imports.

The formula for the income approach is as follows:

$$\hbox{GDP} = \hbox{Total Wages + Total Profits +Total Interest + Total Rent + Proprietors income + Taxes}$$

Next is the expenditures approach. The expenditure approach is calculated by the total of a country's expenditures across four major categories. These categories are consumer spending, business investment, government spending, and net exports, which are exports minus imports.The spending measured in the expenditure approach is exclusively for final goods, as intermediate goods are not counted. That means that many different production processes a product goes through do not matter, but only the final cost of the good does. This is done to prevent double counting the value of goods.

The formula for the expenditure approach is the following:

$$\hbox{GDP} = \hbox{C + I + G + NX}$$

$$\hbox{Where:}$$$$\hbox{C = Consumer Spending}$$$$\hbox{I = Business Investment}$$$$\hbox{G = Government Spending}$$$$\hbox{NX = Net Exports (Exports - Imports)}$$

These figures will leave you with what's called the nominal GDP, which is a snapshot of current prices and outputs.

However, real GDP is when inflation is accounted for in price changes. This is done as rising prices without corresponding increases in value will skew one into thinking the economy has grown. The real GDP is a comparison of the prices of a base year to the current year. This important distinction lets economists measure real growth in value rather than inflationary price increases. A GDP deflator is a variable that will adjust the nominal GDP for inflation.

$$\hbox{Real GDP} = \frac{\hbox{Nominal GDP}} {\hbox{GDP Deflator}}$$

For more information on the distinction between nominal and real GDP, read our explanation of it!

Another way to calculate GDP is the value-added approach. This approach, rather than taking the final good, calculates the additional value each step of production adds to a product.

$$\hbox{Value-Added} = \hbox{Sale Price} - \hbox{Cost of Intermediate Goods and Services}$$

$$\hbox{GDP} = \hbox{Sum of Value-Added for All Products and Services in the Economy}$$

Imagine your favorite candy. You buy it in a package for two dollars, but what is the value-added breakdown of that package?

Raw materials such as sugar cane, starch, and palm oil are harvested and processed.The materials are sold to a candy factory for $0.30 (per package). The candy factory processes it into the shape and flavor you know and love. The candy factory sells the candy packages to a distributor for$1.50 (per package).

The distributor provides boxes of candy to be featured in convenience stores.The packages of candy here are sold for $2.00 Following the example, the farmers who grew and harvested the raw materials created$0.30 of value.

The candy factory that processed the materials sold them for $1.50. However, we must subtract the$0.30 value they paid for.

Therefore, the candy factory added a $1.20 value. Finally, the distributor handled the shipping and logistics to provide the candy at the consumer's convenience for$2.00, minus the $1.50 they paid, so the distributor added$0.50 of value.

$$\hbox{Value-Added} = \ 0.30+1.20+0.50=2$$

In the example above, the value-added approach isolates each step of the production process and the value they provide. However, as you can see, it is the same as the final good price.

## National Output Examples

To have an example of national output, we will use the expenditure approach to calculate the actual GDP for the United States in 2021, Quarter 1.

We will look at all the productive categories of an economy and use them to calculate the value of all final goods produced.

$$\hbox{GDP} = \hbox{C + I + G + NX}$$

$$\hbox{Where:}$$$$\hbox{C = Consumer Spending}$$$$\hbox{I = Business Investment}$$$$\hbox{G = Government Spending}$$$$\hbox{NX = Net Exports (Exports - Imports)}$$

Using data from the Bureau of Economic Analysis, quarter 1 of 2021.3 (numbers are in billions of US dollars)

$$\hbox{C = 14,439.1}$$$$\hbox{I = 3,752.4}$$$$\hbox{G = 3,831.6}$$$$\hbox{NX = -541.7}$$

$$\hbox{GDP} = \hbox{14,439.1 + 3,752.4 + 3,831.6 + (- 541.7)}$$

$$\hbox{GDP} = \hbox{21,481.4 billion US dollars}$$

The example above can be seen in the graph below in Figure 1. The sharp rise in GDP after 2020 is from the calculations that were done in the example above. Notably, the major dips in GDP occur during massive financial crises, such as the 2008 financial crisis and the 2020 Covid-19 pandemic.

Fig. 1 - US real GDP growth. Source: Bureau of Economic Analysis4

## Measuring Domestic Output and National Income - Key takeaways

• Domestic output refers to the gross domestic product, which is the total productive output of a country.
• National income is the total money earned across all economic actors, from households to businesses and the government.
• There are three ways of calculating GDP: the expenditure approach, the income approach, and the value-added approach.
• Nominal GDP is the current output measured with current prices, whilst real GDP measures the actual growth in productive value by removing inflation from the price changes.

## References

1. GNI United States 2021, https://www.macrotrends.net/countries/USA/united-states/gni-gross-national-income
2. Table 1 Federal reserve economic data, https://fred.stlouisfed.org/release/tables?rid=53&eid=42133
3. Expenditure approach GDP Data, https://apps.bea.gov/iTable/iTable.cfm?reqid=19&step=2#reqid=19&step=2&isuri=1&1921=survey
4. Fig. 1 US Real GDP growth, Bureau of Economic Analysis Table 1.1.1 https://apps.bea.gov/iTable/iTable.cfm?reqid=19&step=2#reqid=19&step=2&isuri=1&1921=survey

## Frequently Asked Questions about Measuring Domestic Output and National Income

Gross domestic product, gross net product, national income are all measures of domestic output. They calculate the total productive value of an economy in a given time frame.

Net domestic product and national income can be calculated in multiple ways. The income approach calculates all incomes of individuals and business. The expenditure approach calculates the final value of goods and services sold. The value-added approached calculates the productive value of each step in a production process.

The three methods are as follows. The income approach calculates all incomes of individuals and business. The expenditure approach calculates the final value of goods and services sold. The value-added approach calculates the productive value of each step in a production process.

The five main categories that national income is characterized by are: compensation of employees, proprietors income, rental income, corporate profits, and net interest.

National income and output are relatively the same, they are both productive measures of an economy. National income focuses on how much the country is paid for its goods and services. The national output is the value of final goods and services produced.

## Final Measuring Domestic Output and National Income Quiz

Question

What is real GDP?

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Answer

Real GDP is GDP that accounts for inflation and has been adjusted to reflect changes in prices.

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What is nominal GDP?

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Answer

Nominal GDP is GDP that has not accounted for inflation and includes its current prices.

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Question

what is GDP?

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Answer

GDP is the value of all final goods and services produced in a specific time period.

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What is the difference between nominal and real GDP?

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nominal GDP does not account for inflation; real GDP does account for inflation.

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How do you calculate nominal GDP?

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Units of Output X Price of Item Per Unit

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How do you calculate real GDP?

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Nominal GDP / Price Index (in hundreds)

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Question

Assume a country only sells laptops. The laptops cost $200 and the country sold 10,000 units. What is nominal GDP? Show answer Answer 2,000,000 Show question Question Assume a country only sells laptops. The laptops cost$10 and the country sold 1,000 units. What is nominal GDP?

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Answer

10,000

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The nominal GDP is 500, and the price index is 150. What is real GDP?

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Answer

333.33

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True or False: real GDP does NOT account for inflation?

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False

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True or False: nominal GDP can also be known as, "current prices"

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True

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True or False: Another way to calculate real GDP is by multiplying the specific year's output by the base year's prices.

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True

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True or False: The formula for nominal GDP is: Nominal GDP = real GDP / price index

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False

Show question

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True or False: The formula for real GDP is: Real GDP = Nominal GDP / Price Index (in hundreds)

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True

Show question

Question

In year 1, there were 30 units of TVs produced at a price of $500. In year 2, there were 50 units of TVs produced at a price of$550. What is real GDP in year 2?

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Answer

25,000.

Show question

Question

Define GDP.

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Answer

GDP is the value, in monetary terms, of an economy's annual total output of goods and services.

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Economists use changes in GDP to measure economic performance.

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True

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There are 5 types of GDP.

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False

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The types of GDP are nominal GDP and _____.

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Real GDP

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Nominal GDP uses constant prices.

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False

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Non-production transactions are not included in the GDP.

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True

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What are non-production transactions?

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Non-production transactions are transactions that are not tied to the production of final goods.

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Real GDP uses current prices.

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False

Show question

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Define nominal GDP.

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Nominal GDP is the monetary value of the total annual goods and services produced in an economy using current prices.

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Define real GDP.

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Real GDP is the monetary value of the total annual goods and services produced in an economy using constant prices.

Show question

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Under the value added approach, GDP is measured by summing up the value added by the government, businesses, and households.

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True

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The expenditures approach of measured GDP puts together all the income on final goods and services in the economy during the year in question.

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False

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Government spending on goods and services is included in the expenditure approach.

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True

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The income approach includes employee compensation.

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True

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Economists use nominal GDP to measure economic growth.

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False

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What is the expenditure approach?

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The expenditure approach is a method used to measure the GDP of a country by taking into account the final value of goods and services.

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The expenditure approach considers ____________.

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Public expenditure

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Expenditure approach calculates _________.

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Nominal GDP

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What is the relationship between the expenditure approach and aggregate demand?

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The expenditure approach, as the name suggests, is focused on the total spending in the economy. The total spending in the economy is also represented by Aggregate Demand. Therefore, the expenditure approach's components are the same as those of aggregate demand.

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What are the four types of spending considered in the expenditure approach?

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1. Consumption
2. Investment
3. Net exports of goods and services
4. Government purchases

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______________ refers to the spending by individuals on final goods and services, including those produced in other countries.

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Personal consumption expenditure

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What are the type of goods included in personal consumption?

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All

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______ goods include short-lived consumer items, such as food, gas, or clothing.

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Non-durable

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Buying the new iPhone 14 is an example of ___________.

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Personal consumption expenditure

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What does investment involve under the expenditure approach?

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Investment involves the purchase of new capital goods (also known as a fixed investment) and the expansion of a company's inventory (also known as inventory investment).

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Which one of the following is not a category of private investment?

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Non-durable goods

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Pfizer spending billions of money on R&D to develop the COVID-19 vaccine is an example of ______________.

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Gross domestic private investment

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What are the categories included in Government purchases?

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1. Spending on goods and services that the government needs to provide public services.
2. Spending on long-lasting public assets like schools and highways.
3. Expenditure on research and development and other activities that add to the economy's stock of knowledge.

Show question

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What is an example of government purchases that the expenditure approach would include?

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An example of government purchases that would be included in the GDP calculation by the expenditure approach is the government buying new software technologies for national defense.

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_____ are defined as the goods and services created within a nation that are sold to buyers outside of that country.

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Exports

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_____ are defined as goods and services produced outside of a country that are sold to buyers of that country.

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Imports

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What is the expenditure approach formula?

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The expenditure approach formula is:

$$GDP=C+I_g+G+X_n$$

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What is the national account?

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The national account refers to accounting for the value of all transactions that take place within an economy.

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What are national accounts?

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National accounts (or other national accounts) in economics refers to the subaccounts that combine to form the GDP.

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What does NDP stand for?

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Net Domestic Product

Show question

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# How to calculate photons/light trajectory under gravity

I'm aware many questions are out there asking similar questions about photons and gravity and I got the basic concept by searching through them. I will just call it light although it may not be the most accurate term for it, but I'm sure what I'm asking is apparent. (feel free to answer with photons, lightlike curves, light, radiation or anything you think is accurate)

My question is as follows:

Consider a light beam, initially having a straight line trajectory. If I put a mass $M$ at distance $r$, trajectory will bend slightly towards the objects side. If I put the mass even closer, light might even hit the mass with following a 'death spiral' path. At one specific mass and distance, one can even make light orbit the mass forever if no other interaction exists (or am I mistaken about that?). How do we anticipate which happens?

To discuss on more specific terms: Say there is a wall at a distance $2d$ from the source, at the middle point $d$ there is an object of mass $M$. We send light with frequency $f$, and the straight line photons would follow without gravity has closest distance of $r$ to the object. How much would light would deviate from its final destination, due to gravity?

• Photons don't have trajectories. A photon is a quantum that is the result of a measurement on a quantum field. – CuriousOne Sep 19 '15 at 19:36
• They move and they are affected by gravity, why wouldn't they have a trajectory? – ozgeneral Sep 19 '15 at 19:39
• Because photons are not particles. You make one measurement on a quantum field in A and find a photon and you do another in B and you find another one but there are no trajectories between A and B. If you have many photons, then you can define a wavefront, though. – CuriousOne Sep 19 '15 at 19:42
• Aren't photons considered both particle and wave? And how can we say "photons are attracted by gravity" if we cannot answer this question? – ozgeneral Sep 19 '15 at 19:46
• Deflection angle is $$\frac{4 M G}{r c^2}$$ – Count Iblis Sep 19 '15 at 20:58

Yes, you can calculate light trajectories in gravitational fields. The equation is $$\left( \frac{1}{r^2} \frac{\mathrm{d}r}{\mathrm{d}\theta} \right)^2 = \frac{1}{b^2} - \left(1- \frac{2M}{r}\right)\frac{1}{r^2}$$

Here, $$r$$ is the distance between the light beam and the massive object, $$\theta$$ is the angle of the light beam, $$M$$ is the mass of the object, and $$b$$ is the impact parameter (the minimum distance between the light beam and mass if there was no bending). See the image below.

Integrating over this equation gives the total deflection of the light during its passage near the massive object. As given in a comment, it is:

$$\Delta \theta = \frac{4MG}{Rc^2}.$$

Deriving these equations requires some very basic General Relativity. You can find a derivation on this lovely website by Christian Magnan, who cites the book "Exploring Black Holes, Introduction to General Relativity" by Edwin F. Taylor and John Archibald Wheeler.

• I don't understand all this pedantry in the comments about the use of the term "photons". In the geometric optics regime (approximation), you can call them photons, light pulses, light beams, or wave packets. You could even consider massive particles moving at 99.99% of the speed of light (neutrinos?). All of these names are the same thing in this regime. In the end, you just calculate a curve in space relative to the massive object, from which you can calculate a trajectory (position as a function of time) knowing that the light pulse always moves at c. – WaterMolecule Nov 9 '18 at 21:00

I forget who said it, but it's a good rule of thumb "Mass-energy tells space how to bend, space tells mass-energy how to move."

Space-time has a certain shape. Just as the shortest distance between two points is not a straight line on the surface of a globe is not a straight line, the shortest distance between two points in space-time is also not a straight line under the influence of gravity .

The shortest distance path on a surface is called a geodesic, whatever the shape of the surface. So a great circle on a sphere is a geodesic, so is a straight line on the euclidean plane. One is not the same as the other, but they both are the paths an object would take if it is not under the influence of an external force.

Consider the graph of the function $$y=x^2$$. And suppose you travel along that graph at a constant speed. $$y=x^2$$ $$\dot{x}^2+\dot{y}^2=u^2.$$

From this it follows that:

$$\dot{x}=\frac{u}{\sqrt{1+4x^2}}$$ $$\dot{y}=\frac{2xu}{\sqrt{1+4x^2}}$$

So, $$\ddot{x}=\frac{4xu}{(1+4x^2)^\frac{3}{2}}$$

$$\ddot{y}=\frac{-2u}{(1+4x^2)^\frac{3}{2}}$$

It's clear that despite maintaining a constant speed, there is a changing velocity, an acceleration since the second derivatives with respect to time are non-zero. But acceleration in the absence of an applied net force is a pseudo-force.

Now the curvature of a curve is $$k=\frac{\frac{d^2y}{dx^2}}{(1+(\frac{dy}{dx})^2)^\frac{3}{2}}=\frac{2}{(1+4x^2)^\frac{3}{2}}$$ in the case of our curve here.

Notice that $$k$$ divides both acceleration terms. This is no coincidence.

Let $$\frac{d\vec{s}}{dt}$$ be velocity along the path. By definition, curvature is :

$$k=\frac{d\theta}{ds}$$ where $$\theta$$ is $$tan^-1(dy/dx)$$.

Another definition of curvature is $$\frac{d\vec{T}}{dt}$$, where $$\vec{T}$$ is the unit tangent vector to the curve, i.e. $$\vec{T}=\frac{d\vec{s}}{ds}=\frac{d\vec{s}/dt}{ds/dt}$$

Now $$ds/dt=u$$ and we are holding $$u$$ constant, so $$du/dt=0$$.

So $$\frac{d}{dt}(\frac{d\vec{s}/dt}{ds/dt})=\frac{1}{u}\frac{d^2\vec{s}}{dt^2}=\vec{k}$$

So we expect the acceleration to be the product of the constant velocity and the curvature.

Notice from this, if we have no curvature, we have no acceleration. Zero curvature implies a straight line and no pseudo-force.

Curvature has a broader definition in 4d space-time.

The equation of geodesics becomes:

$$\frac{d^2x^a}{dt^2}=\sum_{\mu=0}^3\sum_{\nu=0}^3-\Gamma_{\mu\nu}^a\frac{dx^\mu}{dt}\frac{dx^\nu}{dt}$$

Here $$x^a$$ means the $$a_{th}$$ component of path $$\vec{s}$$.

$$\Gamma^a_{\mu\nu}$$ is the Christofel Symbol of the Second Kind.

Notice again we have a second time derivative being the product of some first time derivative. As it happens the Christofel Symbol has a close relation to curvature.

The Stress-Energy Tensor, roughly equivalent to the mass distribution of Newtonian Gravity, dictates curvature of your region of space which can be expressed in terms of the Christofel Symbols.

If we know the Christofel Symobols, then the geodesic equation will tell us the path particles will take in the absence of an applied force. It will tell us how photons move.

Notice the geodesic equation makes no explicit reference to some mass energy distribution. The Christofel symbols dictate our geometry. All we need to know is the geometry to find our paths.

So that quote above actually breaks up the problem of General Relativity. Understanding geometry of space time, and understanding how that is manipulated by mass energy. For the sake of theoretical work, they can be given arbitrarily to see what paths result.

So curvature determines which paths exist in the first place and and light "selects" the Least Time Path according to Fermat's Principle.

format: plain


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Covid-19 model scenarios for UK

Social distancing slows down the spread of Covid-19 by reducing the reproduction factor $R_0$ . If $R_0$ is reduced below 1,  then the number of cases and deaths will begin to fall, but how effective is this strategy long term and how can we escape from a lockdown once it has been  imposed. I have looked a bit into this using my “Mickey Mouse” SIR model. Interestingly enough it gives similar results to that of Neil Fergusen if I use the same parameters (R=2.5 etc).

First here is the simulation for the UK under the “take it on the chin” scenario.

“Take it on the chin”. The epidemic is over after ~ 3 months but at the cost of half a million deaths !

These results assume a death rate of 1% and an effective $R_0$ of 2.5. Half a million deaths is unacceptable for any government which probably explains why the lockdown measures were introduced, once the Imperial study was published. The goal of a lockdown is to reduce the reproduction rate $R_0$ to below 1 by decreasing the contact rate $\beta$ (social interaction) by about 70%. Once successful this lockdown would then result in the following distribution assuming the measures are introduced after ~40 days into the epidemic (mid March) and ended after 100 days.

Effects of a 3 month long lockdown followed by a return to near normal

It takes about 3 months for the lockdown to reduce the number of active infections to a low level (<10%). However once the restrictions are lifted the epidemic quickly takes off again, which then triggers another full lockdown. If such action is taken quick enough then the epidemic can finally be stopped as shown below.

Stop-Go scenario: A lockdown and release followed by a second lockdown a month later.

This last scenario seems finally to work in reducing the infected cases to a very low level. However the final number of deaths is still large ~92000. The damage to the UK economy of a 3 month shutdown followed by a second emergency shorter shutdown would be  enormous. Could the UK ever recover from such an economic shock and would  the UK public be willing to accept such a 7 month period of drastic stop-go action ?

These scenarios may be far too pessimistic. Two recent studies of antibodies in sample populations seem to show that the infection rate (15%) is currently much higher than thought, and consequently that the death rate could be as low as 0.3% (see my comment below & here). That would reduce the predicted death tolls by a factor of 3.  Furthermore one would hope that more effective treatments will be found to reduce the death rate even further. An effective vaccine would eliminate the risk completely.

My raw python code can be found here

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
This entry was posted in Health and tagged . Bookmark the permalink.

12 Responses to Covid-19 model scenarios for UK

1. Andrew Carey says:

Does your code allow for R0 changing over time : initially the biggest most mobile most social people spreading the virus are most likely to get it and would presumably be excluded from spreading it again.

• Clive Best says:

Yes it is easy to change that over time. In fact that is all I essentially do for lockdowns by changing beta – the social interaction parameter with time. I should probably tidy up the code and make all parameters configurable in a file. Right now I just hack the code.

2. Bryce F Payne says:

Clive, you state,
“Two recent studies of antibodies in sample populations seem to show that the infection rate (15%) is currently much higher than thought…”
Which studies were those?

• Clive Best says:

The first was a German study in Gangelt which was heavily effected with Covid-19. 14% of 500 randomly sampled people were antibody positive which implied and infection to mortality rate -> 0.37%.

See here

The second was in the town of Robbio, Lombardia which likewise found an infection rate of 14%. My guess is that the same percentage might now apply to London, most of whom were asymptotic.

• Hello Clive, the random testing suggests infection rates of between 10 and 15%. I can’t see how 0.1% of UK population could infect half the Scottish care homes. It must be that that the virus is more visible there.

• Clive Best says:

Hi Iain,

You probably only need one infectious care worker per care home to eventually infect all the residents. So if you have say 70 residents who meet up daily in a communal lounge with 50 staff (cooks, cleaners, helpers, nurses etc), then I think it probably is likely that half care homes get infected with Covid-19 even at a 10% rate.

• ok to infect 50% of care homes you would need 0.7% of 70 care workers to be infected.

3. Robert Johnson-Taylor says:

The trouble with all these models they are too narrowly focused. It is necessary to consider the deaths caused by enforced lock down; families forced to remain in close proximity for protracted periods of time, not only result in assults, but also murders and suicides. The elderly living alone without adequate, if any support, resulting in death by neglect. The mentally ill not capable of being left to support them selfes. The UK government is allready building up body recovery resources for post lockdown, and are being told this is going to be very unpleasent. But the real killer, the biggest killer of all is poverty. Employment that once existed will have gone, together with their markets. This economic reality will be inter-generational

• Clive Best says:

You are absolutely right. The long term effects of any extended lockdown are huge and unknown. Perhaps governments have over-reacted – even Trump has been forced into a lockdown against his instincts. I can’t see any way out of this unless we can first find an effective treatment to cut deaths, and later develop a vaccine. Until humans build an immunity to the virus through vaccine or disease we can only suppress infections in the short term, but as you say at huge social and economic costs.

• Robert Johnson-Taylor says:

I have seen posts, from reputable sources, that there is a growing body of evidence that COVID19 is a man made virus. no I’m not a conspiracy theorist. The Chinese government produced it, and it escaped, not a deliberate act.

4. Clive Best says:

Hong Kong closed the borders early and has an efficient testing, tracking and quarantine program. The first case occurred on 23 January and since then there have been as of today (13th April) only 1004 cases with just 4 deaths. This implies a death rate of 0.4%

If this applies globally then for the UK the above estimates would give

“Take it on the chin” : 230,000 deaths
“Stop Go” : 48000 deaths but still a small number (140) left infected risking a further increase.

5. mesocyclone says:

A strategy that a lot of countries are going for is to use the strong social distancing to knock the number of cases down to a level where traditional means – testing, and contact tracing with individual quarantine – can be used after that, with a lot less social distancing. Then… keep the levels low until a vaccine is available. In that strategy, Rt is kept lower than R0 by some distancing, but mostly by containing any outbreak before the numbers get too big.

That could not be used at first because too much was unknown, plus it takes time to scale up testing and contact tracing logistics.

BTW, I’m not an epidemic modeler, but my reading suggests that Rt (short for R(t)) is used for the reproduction number that diverges in time (for whatever reason) from R0.


text:
# How to have multiple citations entries using footcite?

I'm using Biblatex to put my references in footnotes. When I previously used \cite{} to insert references, I could just separate the references with a comma to put in multiple references: \cite{ref1, ref2} would result in blah blah blah [1,2]; and \cite{c1,c2,c3} would result in [1-3]

I tried this with \autocite, but it doesn't seem to work. Is there an easy way to get this to work?

Here is an example:

\documentclass{article}
\usepackage[style=verbose,backend=bibtex, autocite=footnote]{biblatex}

\begin{filecontents*}{bibliography.bib}
@article{c1,
author = {Neil, Daniel and Liu, Shih-Chii},
journal = {IEEE Transactions on VLSI Systems},
title = {{Minitaur, an Event-Driven FPGA-Based Spiking Network Accelerator}},
year = {2014}
}

@book{c2,
title={Event-Based Neuromorphic Systems},
author={Liu, Shih-Chii and others},
year={2015},
publisher={John Wiley \& Sons}
}
@inproceedings{c3,
title = {Personalized Speech Recognition On Mobile Devices},
author  = {Ian McGraw and others},
year  = 2016,
booktitle = {ICASSP}
}
\end{filecontents*}

\bibliography{bibliography.bib}

\begin{document}
I would like the footnote citations~\autocites{c1,c2,c3} to look $like^{1-3}$, but I got 1.
\end{document}

• Mhhh, you are just getting citations in a footnote here, you are not using a numeric style. The 1 is the footnote number not the label of any of your citations. – moewe Nov 5 '16 at 13:43
• Anyway to work around this? e.g. footcite? – user48564 Nov 5 '16 at 21:37
• Not really, \autocite is just \footcite here. Why exactly do you want multiple footnote numbers here? Do you have a numeric bibliography? Note that you simply have a footnote with the bibliographic info here. – moewe Nov 5 '16 at 21:48
• I would like to have a numeric bibliography where the citation numbers are displayed like footnote numbers in the main texts. Thanks for the help. – user48564 Nov 7 '16 at 11:49
• But how would you then distinguish real footnotes from citations? Would you still want to see the citation in the footnotes like in your example? – moewe Nov 7 '16 at 20:21


text:
one bridge further

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , on June 30, 2020 by xi'an

Jackie Wong, Jon Forster (Warwick) and Peter Smith have just published a paper in Statistics & Computing on bridge sampling bias and improvement by splitting.

“… known to be asymptotically unbiased, bridge sampling technique produces biased estimates in practical usage for small to moderate sample sizes (…) the estimator yields positive bias that worsens with increasing distance between the two distributions. The second type of bias arises when the approximation density is determined from the posterior samples using the method of moments, resulting in a systematic underestimation of the normalizing constant.”

Recall that bridge sampling is based on a double trick with two samples x and y from two (unnormalised) densities f and g that are interverted in a ratio

$m \sum_{i=1}^n g(x_i)\omega(x_i) \Big/ n \sum_{i=1}^m f(y_i)\omega(y_i)$

of unbiased estimators of the inverse normalising constants. Hence biased. The more the less similar these two densities are. Special cases for ω include importance sampling [unbiased] and reciprocal importance sampling. Since the optimal version of the bridge weight ω is the inverse of the mixture of f and g, it makes me wonder at the performance of using both samples top and bottom, since as an aggregated sample, they also come from the mixture, as in Owen & Zhou (2000) multiple importance sampler. However, a quick try with a positive Normal versus an Exponential with rate 2 does not show an improvement in using both samples top and bottom (even when using the perfectly normalised versions)

morc=(sum(f(y)/(nx*dnorm(y)+ny*dexp(y,2)))+
sum(f(x)/(nx*dnorm(x)+ny*dexp(x,2))))/(
sum(g(x)/(nx*dnorm(x)+ny*dexp(x,2)))+
sum(g(y)/(nx*dnorm(y)+ny*dexp(y,2))))

at least in terms of bias… Surprisingly (!) the bias almost vanishes for very different samples sizes either in favour of f or in favour of g. This may be a form of genuine defensive sampling, who knows?! At the very least, this ensures a finite variance for all weights. (The splitting approach introduced in the paper is a natural solution to create independence between the first sample and the second density. This reminded me of our two parallel chains in AMIS.)

nested sampling via SMC

Posted in Books, pictures, Statistics with tags , , , , , , , , , , , , on April 2, 2020 by xi'an

“We show that by implementing a special type of [sequential Monte Carlo] sampler that takes two im-portance sampling paths at each iteration, one obtains an analogous SMC method to [nested sampling] that resolves its main theoretical and practical issues.”

A paper by Queenslander Robert Salomone, Leah South, Chris Drovandi and Dirk Kroese that I had missed (and recovered by Grégoire after we discussed this possibility with our Master students). On using SMC in nested sampling. What are the difficulties mentioned in the above quote?

1. Dependence between the simulated samples, since only the offending particle is moved by one or several MCMC steps. (And MultiNest is not a foolproof solution.)
2. The error due to quadrature is hard to evaluate, with parallelised versions aggravating the error.
3. There is a truncation error due to the stopping rule when the exact maximum of the likelihood function is unknown.

Not mentioning the Monte Carlo error, of course, which should remain at the √n level.

“Nested Sampling is a special type of adaptive SMC algorithm, where weights are assigned in a suboptimal way.”

The above remark is somewhat obvious for a fixed sequence of likelihood levels and a set of particles at each (ring) level. moved by a Markov kernel with the right stationary target. Constrained to move within the ring, which may prove delicate in complex settings. Such a non-adaptive version is however not realistic and hence both the level sets and the stopping rule need be selected from the existing simulation, respectively as a quantile of the observed likelihood and as a failure to modify the evidence approximation, an adaptation that is a Catch 22! as we already found in the AMIS paper.  (AMIS stands for adaptive mixture importance sampling.) To escape the quandary, the authors use both an auxiliary variable (to avoid atoms) and two importance sampling sequences (as in AMIS). And only a single particle with non-zero incremental weight for the (upper level) target. As the full details are a bit fuzzy to me, I hope I can experiment with my (quarantined) students on the full implementation of the method.

“Such cases asides, the question whether SMC is preferable using the TA or NS approach is really one of whether it is preferable to sample (relatively) easy distributions subject to a constraint or to sample potentially difficult distributions.”

A question (why not regular SMC?) I was indeed considering until coming to the conclusion section but did not find it treated in the paper. There is little discussion on the computing requirements either, as it seems the method is more time-consuming than a regular nested sample. (On the personal side,  I appreciated very much their “special thanks to Christian Robert, whose many blog posts on NS helped influence this work, and played a large partin inspiring it.”)

Why do we draw parameters to draw from a marginal distribution that does not contain the parameters?

Posted in Statistics with tags , , , , , , , on November 3, 2019 by xi'an

A revealing question on X validated of a simulation concept students (and others) have trouble gripping with. Namely using auxiliary variates to simulate from a marginal distribution, since these auxiliary variables are later dismissed and hence appear to them (students) of no use at all. Even after being exposed to the accept-reject algorithm. Or to multiple importance sampling. In the sense that a realisation of a random variable can be associated with a whole series of densities in an importance weight, all of them being valid (but some more equal than others!).

revisiting the balance heuristic

Posted in Statistics with tags , , , , , , , on October 24, 2019 by xi'an

Last August, Felipe Medina-Aguayo (a former student at Warwick) and Richard Everitt (who has now joined Warwick) arXived a paper on multiple importance sampling (for normalising constants) that goes “exploring some improvements and variations of the balance heuristic via a novel extended-space representation of the estimator, leading to straightforward annealing schemes for variance reduction purposes”, with the interesting side remark that Rao-Blackwellisation may prove sub-optimal when there are many terms in the proposal family, in the sense that not every term in the mixture gets sampled. As already noticed by Victor Elvira and co-authors, getting rid of the components that are not used being an improvement without inducing a bias. The paper also notices that the loss due to using sample sizes rather than expected sample sizes is of second order, compared with the variance of the compared estimators. It further relates to a completion or auxiliary perspective that reminds me of the approaches we adopted in the population Monte Carlo papers and in the vanilla Rao-Blackwellisation paper. But it somewhat diverges from this literature when entering a simulated annealing perspective, in that the importance distributions it considers are freely chosen as powers of a generic target. It is quite surprising that, despite the normalising weights being unknown, a simulated annealing approach produces an unbiased estimator of the initial normalising constant. While another surprise therein is that the extended target associated to their balance heuristic does not admit the right density as marginal but preserves the same normalising constant… (This paper will be presented at BayesComp 2020.)

a new rule for adaptive importance sampling

Posted in Books, Statistics with tags , , , , , , , , , on March 5, 2019 by xi'an

Art Owen and Yi Zhou have arXived a short paper on the combination of importance sampling estimators. Which connects somehow with the talk about multiple estimators I gave at ESM last year in Helsinki. And our earlier AMIS combination. The paper however makes two important assumptions to reach optimal weighting, which is inversely proportional to the variance:

1. the estimators are uncorrelated if dependent;
2. the variance of the k-th estimator is of order a (negative) power of k.

The later is puzzling when considering a series of estimators, in that k appears to act as a sample size (as in AMIS), the power is usually unknown but also there is no reason for the power to be the same for all estimators. The authors propose to use ½ as the default, both because this is the standard Monte Carlo rate and because the loss in variance is then minimal, being 12% larger.

As an aside, Art Owen also wrote an invited discussion “the unreasonable effectiveness of Monte Carlo” of ” Probabilistic Integration: A Role in Statistical Computation?” by François-Xavier Briol, Chris  Oates, Mark Girolami (Warwick), Michael Osborne and Deni Sejdinovic, to appear in Statistical Science, discussion that contains a wealth of smart and enlightening remarks. Like the analogy between pseudo-random number generators [which work unreasonably well!] vs true random numbers and Bayesian numerical integration versus non-random functions. Or the role of advanced bootstrapping when assessing the variability of Monte Carlo estimates (citing a paper of his from 1992). Also pointing out at an intriguing MCMC paper by  Michael Lavine and Jim Hodges to appear in The American Statistician.


text:
## Intermediate Algebra for College Students (7th Edition)

$x^2-9$
$x+3=x+3$, $x^2-9=(x+3)(x-3)$, so the GCF is $x+3$ and the terms that are "not common" are $x-3$ so LCD$=(x+3)(x-3)=x^2-9$