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# Developing economies in international trade

SDG indicators
Target 17.11: Significantly increase the exports of developing countries, in particular with a view to doubling the least developed countries’ share of global exports by 2020.
Indicator 17.11.1: Developing countries’ and least developed countries’ share of global exports (Tier I)

Target 8.9: By 2030, devise and implement policies to promote sustainable tourism that creates jobs and promotes local culture and products.
Indicator 8.9.1: Tourism direct GDP as a proportion of total GDP and in growth rate (Tier II)

Trade is recognized as a key factor for the 2030 Agenda, including poverty reduction and economic growth (Tipping and Wolfe, 2016). SDG target 17.11 aims to significantly increase the exports of developing countries, and in particular with a view to doubling the LDC’s share in global exports by 2020. Reaching this target is likely to be implausible. As will be seen below, there has not been a substantial increase in the share of exports for LDCs or for developing economies in general since 2012. The COVID-19 pandemic poses additional challenges for developing economies in fulfilling not only international trade goals but also various other SDGs.

## COVID-19 pandemic poses a significant challenge to the world trade

World merchandise exports rose by just over 50 per cent over the ten years from 2009 to 2019, reaching US$18.9 trillion in 2019. Nevertheless, this was also a three per cent decline on 2018. In 2017-2018, exports showed signs of recovery after more sluggish performances in 2015 and 2016. In 2019, global services trade was valued at US$6.1 trillion, recording a slight increase of two per cent on 2018, and of almost 70 per cent on ten years earlier (UNCTAD, 2020a).

2020 got off to a rocky start due to the COVID-19 pandemic. Preliminary UNCTAD-WTO estimates (UNCTAD, 2020a) for the first quarter of 2020 show a decline of 2.8 per cent in world merchandise exports on the corresponding quarter in 2019. The seasonally adjusted figures enable comparison with the previous quarter and show a drop of 2.0 per cent for world export volume indices. Most of the impact of COVID-related confinement measures affected global trade during the second quarter of the year, for which UNCTAD estimates a decline of 26.9 per cent from the previous quarter (UNCTAD, 2020b). UNCTAD also forecasts growth in merchandise trade for the year as a whole at -20 per cent.

## Trade openness of developing economies

As shown in figure 1, developing and developed economies’ trade openness indices are converging. LDCs’ trade openness, i.e. the ratio of exports and imports to GDP, has been consistently lower than in other developing economies. The global dip of 2009, associated with the financial crisis, was followed by a short recovery in trade openness for developing economies, but since 2011 their trade openness has drifted downward, bouncing back only slightly after 2016.

From 2014 to 2017, LDCs experienced a persistent decline in trade openness with the index dropping from 59 to 47 per cent (see figure 1). In 2018 this number rose to 53 per cent. This highlights the increasing influence of trade in LDCs’ economies, which might exacerbate the challenges of coping with the economic impacts of the COVID-19 pandemic.

(Percentage)
Notes: This index measures the relative importance of international trade in goods and services relative to the domestic economic output of an economy. It compares the sum of exports and imports to GDP.

## Current drift of trade in developing economies

After the 2008 global financial crisis and the more recent trade downturn in 2014-2016, developing economies have seen a strong recovery since 2017. Goods trade increased at annual growth rates of 11.7 and 10.0 per cent in 2017 and 2018, respectively (figure 2). Trade in services grew by 9.0 in 2017 and 11.6 per cent in 2018. While trade in services in developing countries continued to grow by 2.7 per cent in 2019, trade in goods decreased by 3.5 per cent.

In 2018, total exports of goods and services reached US$10.4 trillion and amounted to US$10.2 trillion in 2019. Thus developing economies’ trade finally exceeded US$10 trillion, a level last achieved in 2014. Their trade has increased by almost 15 per cent since 2015, the year the 2030 Agenda began. In 2020, global trade is expected to fall as the COVID-19 pandemic disrupts economic activity around the world. These disruptions will have profound implications for the most vulnerable economies, including developing economies and LDCs (UNCTAD, 2020c). Figure 2. Trends of goods and services trade in developing economies ## Developing countries performance with respect to SDG 17.11.1 The evaluation of progress towards SDG target 17.11, to significantly increase the exports of developing countries, and to double the LDCs’ share of global exports by 2020, requires a choice of a baseline year. According to the IAEG-SDGs (United Nations, 2019), the default baseline year for each indicator should be 2015. However, some exceptions may be necessary to allow a longer review of trends. Five years is hardly enough time to double the LDCs’ share of global exports. Therefore, for SDG 17.11.1, an earlier baseline year is arguably more appropriate. Yet, whatever the baseline is for the past 20 years, developing countries’ share of global exports has not increased significantly, nor has LDCs’ share doubled. However, at a country level, performances differ and will vary depending on the chosen baseline year (see map 1). The baseline selected for MDGs, for instance, was 1990 – ten years before their adoption in 2000. This gave time for countries to achieve progress and allowed for a more ambitious agenda. If a similar approach was applied to the SDGs, a comparable baseline (ten years prior to adoption) would be 2005. Table 1. Growth rates of exports shares in global trade (Different baselines scenario, in percentage) Alternative baselines Share of global trade (percentage) 2019 Change from baseline (percentage points) Group of economiesMeasure20052010201520192005-20192010-20192015-2019 LDCsService exports0.460.610.740.770.300.150.03 Goods exports0.721.000.890.960.24-0.040.07 Total exports0.680.920.860.910.23-0.010.06 Developing economiesService exports23.1327.6529.5230.046.912.390.52 Goods exports36.2642.0844.7944.268.002.19-0.53 Total exports34.6039.1141.2640.796.191.68-0.47 Another measurement issue to consider is the composition of LDCs. Several LDCs are likely to graduate from this status in the coming years. According to the UN Department of Economic and Social Affairs (2019), Vanuatu is expected to graduate in 2020, and several others will follow after the end of the target year, 2020. MacFeely (2020) has discussed the implications of the changing group composition for assessing progress towards the SDG target. Will the rates of change be calculated using the original composition of LDCs or developing economies at the baseline (say 2010/2011 or 2015), or the group as it will be composed in 2020? These choices leave considerable room for the interpretation of success. Some soon-to-graduate countries have only a marginal contribution on the group performance, and whether they are included or not will have little impact, whereas the weight of some other countries is considerable, like that of Bangladesh (see map 1) and will have a significant impact on the performance of the group as a whole. For the reasons outlined above, the 2010 has been selected as an appropriate baseline year for the scenario discussed in this chapter. Data for additional years are available also. Map 1 shows developing countries’ share of global trade goods exports as well as services exports by country. Map 1. Developing countries’ share of global trade of good and services (Percentage of total trade) Several countries doubled their share of global trade from 2010 to 2019. Viet Nam’s share of world exports of goods grew from 0.47 per cent in 2010 to 1.4 per cent in 2019. Its share of world exports of service also grew from 0.19 per cent to 0.45 per cent. Thailand almost doubled their share of world services exports (from 0.87 to 1.34 per cent), and United Arab Emirates multiplied by 4 their share of service exports (from 0.3 to 1.2 per cent). Bangladesh almost doubled their share of total services exports as well as total goods exports (from 0.13 to 0.20 per cent for goods and from 0.06 to 0.1 for services). ## Developing economies keeping pace with world exports Over the last two decades, developing economies have recorded a notable increase in their share of world trade. Though the value of exports of goods and services from developing countries has increased notably since 2000, since 2012 this growth has no longer outpaced the developed world. Developing countries’ share of global exports of goods and services has risen from 29.7 per cent in 2000 to 41.4 per cent in 2012 but has stagnated ever since. If the baseline selected is 2015, there would be a 0.47 percentage point decrease by 2019. From 2010, developing economies’ share of global trade has increased by 1.68 percentage points and, from 2005, 6.19 percentage points. As far as exports of goods is concerned, developing economies’ share in world exports of goods has plateaued at just above 44 per cent since 2012 (see figure 3). In 2019, developing economies’ share of world services exports (US$6.1 trillion) was 30 per cent (US$1.83 trillion). The highest share of world services exports was recorded by developing Asia at more than 24 per cent. The top three services exporters are China (4.6 per cent), India (3.4 per cent) and Singapore (3.5 per cent). They account for more than 40 per cent of developing economies’ services exports. Figure 3. Developing economies' shares of global exports (SDG 17.11.1) of goods and services (Percentage) Source: UNCTAD (2020a). Note: Statistics on trade in services are preliminary, annual estimates based on the most recent quarterly figures (BPM6). Statistics on trade in goods are estimates based on Comtrade, international and national sources. LDCs’ are a small player in world trade with a 0.91 per cent share in 2019. The 2030 Agenda sets a target to double LDCs’ share in global exports by 2020. LDCs’ share of global exports of goods and services was 0.92 per cent in 2010, slightly above the 2019 level. Taking 2005 as the base, their share in global trade increased by 0.23 percentage points from 0.68 per cent to 0.91 in 2019. LDCs have a long way to go before doubling their share. In 2019, the value of merchandise exports from LDCs was US$180.9 billion, accounting for about one per cent of world exports. Their share in world merchandise exports almost doubled from 0.54 per cent in 2000 (US$35 billion) to over one per cent in 2011-2013 (see figure 4). Since then, this trend has reversed slightly, and it seems unlikely that LDCs will achieve the target in 2020. Figure 4. LDCs' share of global exports (SDG 17.11.1) of goods and services (Percentage) The key driver of export growth over this period was the massive rise in the price of fuels, ores and metals, reflecting high demand in developing countries, most notably China. With 2005 taken as the baseline, the growth is more notable, 0.3 percentage points from 0.5 per cent to 0.8 in 2019 (UNCTAD, 2016). ## China, EU28 and the United States of America are the top trading partners of LDCs In 2018, developing economies shipped most of their exports to the United States of America (US$1.4 trillion), China (US$1.1 trillion) and other Asian economies. For LDCs, the top export destination was China (US$36 billion). LDCs in Africa and Haiti delivered goods worth US$25.7 billion to China, more than to any other economy in the world (see figure 5). LDC exports in Asia were oriented towards China and the United States of America in 2018. The importance of the European Union as a trading partner for LDCs in Asia has increased significantly since the turn of the century, with exports reaching US$49.9 billion in 2018. Intra-regional trade is also high for LDCs from East Asia and the Pacific, and low but rising for LDCs from most other regions.

As merchandise exports of LDCs are concentrated in a few markets, including those worst affected by the COVID-19 health crisis (China, France, Germany, the United States of America), it makes them even more vulnerable to decline in demand in these countries. At individual country level, LDCs are even more exposed to COVID-19 related economic disruptions. For example, in 2018, Angola exported around 57 per cent of its merchandise to China, Benin around 41 per cent to India, Burkina Faso around 54 per cent to Switzerland, Haiti around 82 per cent to the United States of America and Rwanda around 65 per cent to the United Arab Emirates (WTO, 2020b).

Figure 5. Top 5 partners for LDCs in merchandise exports
(Ranked by 2018, billions of USD)

Finally, it's worth noting that the current health crisis has also challenged developing economies to boost their intra-regional trade and strengthen international trade agreements to harmonize their trade-related regulations, customs controls, and reduce both tariff and non-tariff barriers (see New protectionism versus inclusive trade).

## Developing economies’ trade hit by the downswing of the Chinese economy

The coronavirus pandemic has instigated a global economic downturn the likes of which the world has not experienced since the Great Depression. GDP in the world's second largest economy – China, fell by 6.8 per cent year-on-year between in January-March (WEF, 2020). In the first quarter of 2020, China’s exports and imports dropped sharply in volume terms compared to the previous quarter, by 21 per cent and 11. 5 per cent, respectively (UNCTAD, 2020a). The economic consequences of the economic downturn in China were quickly felt in other economies.

China is a major player in international trade as a manufacturer and exporter of consumer products, and as a key supplier of intermediate inputs for manufacturing companies globally. Today about 20 per cent of global trade in manufactured intermediate products originate in China (up from 4 per cent in 2002). UNCTAD (2020d) has analysed the UN Comtrade dataset for about 200 countries and 13 manufacturing sectors to measure each country’s and industry’s integration with the Chinese economy using the GLI of intra-industry trade.

According to this analysis, the economic downturn in China will lead to disruptions in GVCs and diverse spill-over effects across economic sectors and countries. The crisis may impact the supply of critical parts from Chinese producers, affecting economic output and trade in any country depending on their dependency of the Chinese economy. These impacts may spread faster than expected due to the common strategy of limited inventories and just-in-time production.

While the European Union, United States of America and Japan would be hit hardest, the Republic of Korea (US$3.8 billion) would be worst hit among developing economies. The GVCs most affected by China span across sectors, especially communication equipment, different types of machinery and electronics as well as automotive industry. For the Republic of Korea, machinery and communication equipment are the sectors most dependent on China. Figure 6. Top 15 developing economies’ estimated trade losses due to the COVID-19 crisis in China (US$ millions)
Note: The impact of other disruptions than those relating to the Chinese economy are not considered.

While Asian developing economies occupy the top of the list of countries most directly linked to China through GVCs, the effects would also be felt in Mexico (US$1.3 billion), Turkey (US$0.4 billion) and Brazil (US$0.08 billion). In Mexico and Brazil, the automotive industry is most directly linked with Chinese value chains, while in Turkey the sector taking the brunt of the Chinese downturn would be textiles and apparel. Considering the wide-ranging impacts, the quick recovery of the Chinese manufacturing (see Towards sustainable industrialization and higher technologies) in March-April 2020 has brought some good news (UNCTAD, 2020e). ## LDCs’ export product mix becoming more diverse The concentration of LDC exports, as measured by the Herfindahl-Hirschman Index1, increased from 2000 to 2008. Since then concentration has gradually declined, converging with patterns typical of developing economies (see figure 7). Developing economies excluding LDCs have followed a similar trend. In other words, their export mix has become more diverse with a slight sustained set-back from 2016 to 2018. Angola, Botswana and Guinea-Bissau are the three developing African countries with the highest concentration index, reaching an index value of more than 0.9, which indicates that their trade is concentrated on a very few products. Angola is highly dependent on trade in petroleum, Botswana on precious stones, and Guinea-Bissau on fruits and nuts.2 In 2018, LDCs as a group recorded an average index of 0.23. Bangladesh had a relatively high export concentration index in 2018 (0.4), the highest index among Asian LDCs (UNCTAD, 2020a). Of developing economies, the product mix of exports is most concentrated in African countries. The export mix is more varied in the developing economies of America, with Guatemala, Mexico and Panama recorded the lowest concentration index in 2018, and Asia, where Turkey, Thailand and China are the top three most diversified countries. It is worth mentioning that diversifying the strategic economic sectors of LDCs, such as food and health sectors, and empowering both productions and services, such as banking, retailing, and public services with high-level of digitization, represent possibilities for these countries to build more resilient and sustainable economies (World Bank, 2020). Figure 7. Product concentration index of exports in LDCs and developing economies (Percentage) Source: UNCTAD (2020a). Notes: An index value closer to one indicates that a country’s exports or imports are highly concentrated in a few products. On the contrary, values closer to zero reflect a more homogeneous distribution of exports or imports among a series of products. The structure of exports by product group has changed significantly in LDCs and developing economies over the last ten years (see figure 8). In 2018, manufactured goods accounted for 36 per cent of total exports in LDCs – a notable increase from 2008. However, only six LDCs—Bangladesh, Cambodia, Haiti, the Gambia, Nepal and Lesotho—received more than 50 per cent of their export revenue from exporting manufactured goods in 2018. Fuels formed the second largest product group in 2018 (27 per cent), while in 2008 they accounted for over half of the exports. The share of ores, metals, precious stones and non-monetary gold increased from almost 12 per cent to 20 per cent in the ten years from 2008 to 2018. The proportion of food items in exports also increased from eight to almost 12 per cent during the same period. Figure 8. Export structure by product group in LDCs and developing countries (Percentage) Source: UNCTAD (2020a). Notes: For the composition of product groups please refer to UNCTAD (2019b). In 2018, manufactured goods accounted for about 70 per cent of total merchandise exports from developing economies – almost as much as from developed economies. The share of fuels has reduced from almost 27 per cent in 2008 to 16 per cent in 2018. Food continues to be strongly represented in the exports of some economies in South America and Eastern Africa in particular, and ores, metals, precious stones and non-monetary gold in the exports of several Southern and Western African and Central Asian economies. Figure 9. Commodities price index and LDC’s total exports (Proportion) Source: UNCTAD (2020a). Notes: All allocated products: (SITC 0 to 8 + 961 + 971); Primary commodities, precious stones and non-monetary gold: (SITC 0 + 1 + 2 + 3 + 4 + 68 + 667+ 971). LDCs are oriented towards commodity exports, accounting for more than 63 per cent of their goods exports. The periods when LDCs’ exports declined more strongly than world exports (2008-2009 and 2014-2016) coincided with falls in commodity prices. During the period between 2000 and 2019, LDCs recorded a first peak of exports in 2008 with more than US$152 billion followed by a strong decrease caused by the financial crisis in 2008-2009. The second peak was recorded in 2013 with almost US$194 billion. Thus, the global financial crisis of 2008 did not cause sustained declines (even though the commodity prices was connected with the financial crisis). Nevertheless, LDCs’ exports seem to follow commodity price index trends (see figure 9). The decline in commodity prices has caused a more persistent decrease since 2014. The current situation with COVID-19 should play out similarly, as the current decrease of commodity price index (29.3 per cent in March 2020) (UNCTAD, 2020f) will impact LDCs’ exports. LDCs will need to diversify their exports to reduce their exposure to such crises. ## Services exports had been increasing across economies Before services were severely affected by the COVID-19 pandemic, the growth of services exports was a general trend across all economic regions, but mainly benefiting developed economies. In 2019, this group still accounted for 67.7 per cent of all traded services. With US$1.8 trillion worth of services exported in 2019, developing economies took only 30 per cent of the global services market. LDCs’ share amounted to almost 0.8 per cent of total services exports.

2019 recorded an increase of 24 per cent in exports of services compared with 20.7 per cent in 2005 (UNCTAD, 2020a). This trend might be explained by factors, such as the increasing commercialization of intangibles, the larger role of services in global value chains and the gradual liberalization of this sector.

Among broad service categories, travel has the most prominent role in developing economies’ exports. At more than US$578 billion, it accounted for 31.5 per cent of the services supplied internationally by developing economies. Transport is also an important export sector for the developing world, worth US$366 billion in 2019. Grouped together, insurance and financial services, and business and intellectual-property-related services account for US$822.6 billion of developing economies’ exports. Smaller in dollar value than transport and travel, but linked to travel, – exports of personal, cultural and recreational services have been the most dynamic sector in LDCs’ services exports. They grew, on average, by over 13 per cent annually between 2010 and 2019. In the same period, notable annual average increases were recorded for charges for the use of intellectual property, transport and travel services (11.5 per cent, 10 per cent, and 6.8 per cent, respectively). Of the broad services items (Other service sector) which accounts for almost 45 per cent of the total traded services in the region, only construction services saw a downturn in the same period (-4.6 per cent). Figure 10. Annual average growth of services exports by LDCs, by service category, 2010-2019 (Percentage) Source: UNCTAD (2020a). Travel is the only type of service export where LDCs and other developing economies have a revealed comparative advantage3. The revealed comparative advantage of travel services for LDCs reached 1.75 in 2019 and was 1.3 for other developing economies(see figure 11). The value is also slightly greater than 1.34 for transport services. Figure 11. Revealed comparative advantage in service exports, 2019 (Proportion) Source: UNCTAD calculations based on data from UNCTAD (2020a). Notes: The revealed comparative advantage is measured as the proportion of a country group’s exports by service category, divided by the proportion of world exports in each category. A country or region is considered to have a revealed comparative advantage for a product or sector if the index is greater than one. ## Exports of services requiring proximity worst hit in the time of COVID-19 There are four different modes of supply for traded across borders: (mode 1) cross-border trade mainly for services transacted via the internet; (mode 2) consumption abroad covering mainly health and education service for foreigners; (mode 3) commercial presence which is specific to locally-established entities like hotels, banks and construction; and finally (mode 4) movement of natural persons which involve for example foreign IT consultants or health workers4. Figure 12 shows the distribution of global service trade by mode of supply. The data refer to 2017, but give a quite good idea about the mode of supply of services worldwide. More than US$10 billion are exported via mode 1 and 3. Naturally, a big part of services provided by mode 1 continue to be exported in the time of COVID-19, but these services, including telecommunications, computer and information services account for less than 10 per cent of total service exports of developing countries.

The three other modes require proximity between importers and exporters. Thus, the related service sectors will be severely affected and most likely will take longer to recover.

As mentioned above, travel and transport are key sectors in driving developing countries service exports, accounting for more than 50 per cent of total service trade of the group. Those sectors are delivered mainly via mode 2, 3 and 4 and covering services such as education, travel, tourism and associated hotels, and restaurant services, as well as air passenger transport services and construction and other business services that require the movement of skilled and unskilled professionals across borders.

Figure 12. Composition of global services trade by modes of supply, 2017
(Billion US\$)
Source: UNCTAD calculations based on UNWTO (2018).

Travel and transport restrictions due to COVID–19 are likely to negatively affect the trade in services in 2020 . Possible scenarios point to declines of 60 per cent to 80 per cent in international tourist arrivals in 2020. According to CCSA (2020), countries with the highest number of reported cases of COVID-19 (see In focus: COVID-19) account for about 55 and 68 per cent of global inbound and outbound tourism expenditure, respectively. The Joint Report (CCSA, 2020) warns that the effect of the crisis will spill out and be significantly more devastating for countries heavily dependent on tourism.

## Impact of COVID-19 on trade

Baldwin (2020) argues that 2020 will experience a more severe trade turndown than the demand shock of the 2008-2009 crisis, as the COVID-19 crisis creates both a demand and supply shock. From the supply side perspective, production is affected for two reasons, because of reductions in labour supply, and because of disruption to value chains. Countries that rely on equipment and components from regions affected by the virus may experience disruptions in the production process (EIF, 2020).

From the demand side, demand for manufactured goods could fall considerably. During confinement, many shops are closed, and people are reducing shopping in person to avoid social contacts. Workers who are required to stay at home in line with “social distancing” measures tend to prioritize saving over spending, thus, propensity to consume decreases. Secondly, firms that are experiencing disruptions in the production process may decrease their consumption of intermediate goods.

The economies of developing and developed countries are highly interlinked. Exports from developing economies to developed countries (the most likely to be affected severely by COVID-19) accounted for more than 43 per cent of developing economies’ total merchandise trade in 2018, while intra-trade exports accounted for almost 55 per cent. Imports from developed economies to developing economies accounted for more than 30 per cent. Trade in goods and services comprised, on average, around 45 per cent of GDP in SIDS, and up to 30 per cent for LDCs (UNCTAD, 2020a).

Small and vulnerable economies are likely to be hit hard because of their dependence on trade as a driver of economic growth, their small domestic markets and low levels of diversification (see figure 7 above), all of which increase their vulnerability to external shocks – as the global financial crisis demonstrated.

That said, the impact of supply and demand shocks on trade can manifest in different ways depending on the country or region. Economies like China, Europe and the United States of America are mostly affected by direct impacts; the majority of developing countries are mostly affected, as of June 2020, by indirect impacts relating to their level of trade dependency with countries affected by the coronavirus. However, as COVID-19 further spreads to developing countries (see In focus: COVID-19), the direct impacts on those countries are likely to increase.

It is plausible to assume that resource-rich developing countries will be also affected by the strong reduction in commodity prices, for example, petroleum and precious metals (see figure X above), caused by reduced international demand for such goods, and that developed countries have been experiencing a drop in the production of transformed manufactured goods (see Towards sustainable industrialization and higher technologies) (UNIDO, 2020).

Global export is still dominated by goods, with a 76 per cent share in 2019. Exports of goods account for 83 per cent of total exports in developing economies and have become more diversified with manufactured goods representing the largest item of merchandise exports (65 per cent of total goods exports in 2018). LDCs, on the contrary, are highly dependent on exports of commodities, which represent more than 70 per cent of their merchandise exports. High dependence on commodity exports makes most LDCs extremely vulnerable to global shocks, such as the current COVID-19 crisis.

However, the impact of the COVID-19 pandemic will be equally devastating for LDCs and other developing countries that do not rely on commodities as a primary source of their foreign revenues. Non-commodity dependent LDCs, such as Bangladesh, Cambodia and Haiti, rely mostly on low-skilled and labor-intensive manufacturing exports, which are at risk of contracting sharply if global demand for manufacturing exports remains depressed in 2020 and beyond. The lack of sufficiently large domestic demand to absorb excess supply as external demand drops is likely to lead to mass layoffs of the labor force in the manufacturing sector. Much of the exports of these countries rely on intermediate imports from abroad, meaning that if the disruption in global production and supply chains continues, these economies may not be able to procure intermediate production inputs, even if there is demand for their products.

## Tourism makes a significant contribution to sustainable development

One of the most important contributors to international trade in services is tourism. In addition to the direct service itself, tourism has large multiplier effects that extend to the domestic economy. It promotes growth and employment in a multitude of economic sectors, such as transportation, hotels and restaurants, retail trade, financial services and cultural services. It also attracts domestic and foreign investment and promotes the development of the private sector. For this is reason, UNCTAD has recognized that touristic services, if properly harnessed, can become an important engine for inclusive and sustainable economic growth in developing countries (UNCTAD, 2017).

For many developing countries, tourism is one of the most important exports and an essential source of revenue. Figure 10 shows that, on average, tourism contributes to the economy at comparable rates in developing, developed and transition economies. However, for LDCs and especially SIDS, this sector is responsible for a larger share of total economic activity. During 2015-2019, tourism accounted for,on average, 4.4 per cent and 13.7 per cent of LDCs and SIDs GDP , respectively. Moreover, the contribution of tourism to the economy seems to be increasing over time.

Figure 13. Direct contribution of tourism to GDP by country group, average
(Percentage of total GDP)
Source: UNCTAD calculations from World Travel & Tourism Council (2020).
Note: Averages include only countries with available data.

As mentioned above, tourism has a multiplier effect on the domestic economy through several channels. One of these, depicted on map 2, is through its direct contribution to employment creation. In addition to SIDS, many countries in all geographic regions, including South-East Asia (Cambodia, Philippines), North Africa (Tunisia, Morocco), the Caucasus (Georgia), the Americas (Belize, Uruguay, Mexico), Europe (Croatia, Montenegro, Iceland, Greece) and Oceania (New Zealand), benefit greatly from the employment generated across the tourism industries. Overall, current estimates place tourism’s direct contribution to worldwide GDP at 3.3 per cent and to global employment at 3.9 per cent (World Travel & Tourism Council, 2020).5

Map 2. Direct tourism contribution to employment, 2019
(Percentage of global employment)

Despite its increasing economic weight, touristic service supply is still relatively concentrated. More than 45 per cent of all international tourists were still travelling to European countries in 2019. As illustrated in figure 11, other regions of the world received a comparatively small share of international tourist arrivals. This is the case of Oceania, Central Asia, Sub-Saharan Africa and Latin America and the Caribbean, regions where many developing economies are located, including many LDCs. In many regions of the world, tourism still has unexploited potential as a means of development.

Figure 14. International tourist arrivals, distribution by region, 2018
(Percentage)
Source: UNCTAD calculations based on data from UNWTO (2020b).

However, this is gradually changing. Worldwide tourist arrivals increased by almost 50 per cent between 2010 and 2018. While tourists travelling to Europe and Northern America increased by only 41 and 32 per cent, respectively, over the same period they increased by 93 per cent in South and South-East Asia and by a remarkable 243 per cent in Central Asia. The only developing region that did not benefit from this dynamism in tourism was Sub-Saharan Africa, where the number of tourists fell by nine per cent over the period.

## Tourism remains vulnerable to global and regional risks

SDG target 8.9 aims to develop and implement policies to promote sustainable tourism that will result in more jobs and support of local cultures and products. However, even if tourism can bring substantial revenues and economic opportunities, it can also bring challenges for sustainable development. For example, tourism can help finance the preservation of historical and environmental treasures, but if poorly managed could also have the opposite effect (UNCTAD, 2016). Tourists also directly contribute to greenhouse gas emissions and climate change in many ways: through transportation by air, rail, road and sea, and by consumption of goods and services whose production is intensive in energy, water or other resources.

Tourism is a labour-intensive sector that could provide employment for a large share of people, including women and other underrepresented groups. It is also a sector with a high concentration of small and medium enterprises, self-employment and family businesses. For these segments, tourism-related economic activity could provide sustained livelihood opportunities and paths towards poverty reduction for women and local communities in developing countries (UNWTO and ILO, 2014).

However, as revealed by the precipitous decline in international travel and tourism in the aftermath of the COVID-19 outbreak, this is a pro-cyclical sector with high elasticity to global and regional economic trends. In addition, it is very sensitive to perceived security, health and environmental risks. Figure 12 shows the daily evolution of commercial flights during early 2020. We can start to detect a declining trend already at the beginning of the year. However, as more countries installed travel restrictions and encouraged the population to stay home, only a small fraction of scheduled flights were maintained, leading to an unprecedented decline in global flight activity. This fall had a large, direct impact on airlines and the air transportation sector at large. But it also negatively affected businesses and individuals that, directly or indirectly, benefit from providing goods and services to the tourists and business travellers that those cancelled flights would have brought. Although data are not yet available, a similar trend is expected in travel by other modes of transports.

Indeed, recent figures already show a catastrophic year for the sector. Tourist arrivals to Thailand fell by 52 per cent in the first four months of 2020, compared to the same period in 2019 (UNCTAD calculations based on data from Refinitiv (2020). Over the same period, the Republic of Korea recorded a fall of 62 per cent in the number of visitor arrivals (Korea Tourism Organization, 2020). UNWTO (2020a) expects an annual fall of between 60 to 80 per cent in international tourist arrivals in 2020, while OECD (2020) expects a 45 per cent decline in the international tourism economy in their most optimistic scenario and a 70 per cent fall in their most pessimistic. However, there is still a large degree of uncertainty surounding such estimates and the full impact of the coronavirus disease outbreak will only be known once countries start lifting travel restrictions and touristic activity gradually recommences.

These figures show that, while international tourism could provide substantial opportunities for many developing economies, it remains exposed to high global and regional volatility.

Figure 15. Worldwide number of commercial flights, 2020
(Number of flights)

## Notes

1. The Herfindahl-Hirschman Index (HHI) is a measure of market concentration. A higher index value indicates a more concentrated export structure.
2. Products classification refers to three-digit level of SITC Revision 3.
3. The revealed comparative advantage is measured as the proportion of a country group’s exports by service category, divided by the proportion of world exports in each category.
4. Examples of the four Modes of Supply (from the perspective of an “importing” country A) (WTO, 2020a)
• Mode 1: Cross-border
A user in country A receives services from abroad through its telecommunications or postal infrastructure. Such supplies may include consultancy or market research reports, tele-medical advice, distance training, or architectural drawings.
Nationals of A have moved abroad as tourists, students, or patients to consume the respective services.
• Mode 3: Commercial presence
The service is provided within A by a locally-established affiliate, subsidiary, or representative office of a foreign-owned and — controlled company (bank, hotel group, construction company, etc.).
• Mode 4: Movement of natural persons
A foreign national provides a service within A as an independent supplier (e.g., consultant, health worker) or employee of a service supplier (e.g. consultancy firm, hospital, construction company).
5. WTTC also calculates that the total contribution of tourism to the economy. This includes, in addition to the direct impacts, the indirect contribution (tourism-related investment spending, government collective spending and domestic supply chain purchases of goods and services) plus the induced contribution (spending of those directly and indirectly employed by the tourism sector). According to these estimates, the total contribution of tourism is 10.4 per cent of GDP and 9.8 per cent of employment. For details on the methodology of these estimates, see WTTC and Oxford Economics (2018).
6. A country is considered to be export-commodity-dependent when more than 60 per cent of its total merchandise exports are composed of commodities.


text:
# Vectors on a Plane (3D Vectors)

#### Information

A plane is a 2-D body that is used to find vectors perpendicular to certain areas as now we are working in 3-D without this area there would be infinite perpendicular vectors as we have 360° to work with now

For example the grey area below shows the plane that we may need to find perpendicular vectors to

The equation for a plane can be found using the equation below and rplacing the values with the vectors that create the plane

$$\underline{r} = \vec{OA} + λ\vec{AB} + μ\vec{AC}$$

This can be written as:

$$\underline{r} = \underline{a} + λ(\underline{b}-\underline{a}) + μ(\underline{c}-\underline{a})$$

Inputting the values for the vectors and solving allows us to get this in the general equation:

$$ax+by+cz+d=0$$

Find the equation of the plane through A(4,2,0),B(3,1,1) and C(4,-1,1)

Using the equation:

$$\underline{r} = \vec{OA} + λ\vec{AB} + μ\vec{AC}$$

$$\underline{r} = \begin{pmatrix}4\\\ 2\\\ 0\end{pmatrix} + λ\begin{pmatrix}-1\\\ -1\\\ 1\end{pmatrix} + μ\begin{pmatrix}0\\\ -3\\\ 1\end{pmatrix}$$

Now we need to find the x,y and z values for this plane

Solving for x (Equation 1):
x = 4-λ+0μ = 4-λ
λ = 4-x

Solving for y (Equation 2):
y = 2-λ-3μ

Solving for z (Equation 3):
z = 0+λ+μ = λ+μ

Using Equation 1 to replace λ in Equation 2 and 3 we get:
y = 2-(4-x)-3μ = -2+x-3μ
z = (4-x)+μ

Now we need to find a way to remove the μ value which can be done by making the equations have the same μ value and either adding or subtracting them to remove it
To do this we can multiply the z equation by 3 so we have the Equation:

3z = 3(4-x+μ) = 12-3x+3μ

Now we can add these two equations together and we get:

3z+y = 12-3x+3μ-2+x-3μ
3z+y = 10-2x

Rearranging this equation gives us:
2x+y+3z-10 = 0

This is now in the general equation frormat that we discussed earlier:

$$ax+by+cz+d=0$$

#### Question 1

Find the equation of the plane in vector and cartesian form

The normal (perpendicular) to a plane is

$$\begin{pmatrix}1\\\ -5\\\ 8\end{pmatrix}$$

and passes through

$$\begin{pmatrix}3\\\ 2\\\ 7\end{pmatrix}$$

We know from the previous lesson that when two vectors are perpendicular there scalar product is equal to 0 as:
cos90° = 0

This means that the vector r is perpendicualar to the plane (1,-5,8) and we can therefore set their scalar products equal to eachother like so knowing that any remainder will be the value for d (distance) as we already have the values for x,y and z:
(x=1, y=-5 and z=8)

$$r·\begin{pmatrix}1\\\ -5\\\ 8\end{pmatrix}$$

=

$$\begin{pmatrix}3\\\ 2\\\ 7\end{pmatrix}·\begin{pmatrix}1\\\ -5\\\ 8\end{pmatrix}$$

(Vector form of the eqaution is shown below)

$$r·\begin{pmatrix}1\\\ -5\\\ 8\end{pmatrix}$$

=

$$(3×1)+(2×[-5])+(7×8)=49$$

Therefore this means that the distanve of the vector d = 49

To complete the equation we need to write down the value of the whole equation in cartesian form

$$\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}·\begin{pmatrix}1\\\ -5\\\ 8\end{pmatrix}$$

=

$$x-5y+8z=49$$

x - 5y + 8z - 49 = 0

#### Question 2

Write the equation of the plane through (2,1,3), given that the vector (4,5,6) is perpendicular

$$r·\begin{pmatrix}4\\\ 5\\\ 6\end{pmatrix}$$

=

$$\begin{pmatrix}2\\\ 1\\\ 3\end{pmatrix}·\begin{pmatrix}4\\\ 5\\\ 6\end{pmatrix}$$

$$r·\begin{pmatrix}4\\\ 5\\\ 6\end{pmatrix}$$

=

$$(2×4)+(1×5)+(3×6)=31$$

Therefore the distance of the vector is d = 31

$$\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}·\begin{pmatrix}4\\\ 5\\\ 6\end{pmatrix}$$

=

$$4x+5y+6z=31$$

4x + 5y + 6z - 31 = 0

#### Intersection of a line and a plane

Find the point of intersection between the line

The normal (perpendicular) to a plane is

$$r=\begin{pmatrix}2\\\ 3\\\ 4\end{pmatrix}+λ\begin{pmatrix}1\\\ 2\\\ -1\end{pmatrix}$$

with plane

$$5x+y-z=1$$

First we need to replace r with the matrix (x,y,z) in order to find the values of x,y and z

$$\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}$$

=

$$\begin{pmatrix}2\\\ 3\\\ 4\end{pmatrix}+λ\begin{pmatrix}1\\\ 2\\\ -1\end{pmatrix}$$

We can change this equation into 3 different equations by seperating the matrices into:

x = 2 + λ
y = 3 + 2λ
z = 4 - λ

And we know that 5x+y-z=1

Replacing the values of x,y and z with these new values we get the new equation:

5(2+λ) + (3+2λ) - (4-λ) = 1
10+5λ + 3+2λ - 4+λ = 1
8λ = -8
λ = -1

$$\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}$$

=

$$\begin{pmatrix}2\\\ 3\\\ 4\end{pmatrix}+(-1)\begin{pmatrix}1\\\ 2\\\ -1\end{pmatrix}$$

x = 2-1 = 1
y = 3-2 = 1
z = 4+1 = 5

Point of intersection = (1,1,5)

#### Distance of a Point from a Plane

A is the point (7,5,3) and plane π has the equation 3x+2y+z=6
Find the equation of the line through A perpendicular to the plane π

$$r = \begin{pmatrix}7\\\ 5\\\ 3\end{pmatrix}+λ\begin{pmatrix}3\\\ 2\\\ 1\end{pmatrix}$$

Find the point (P) of intersection with the line r and the plane π

$$\begin{pmatrix}x\\\ y\\\ z\end{pmatrix} = \begin{pmatrix}7\\\ 5\\\ 3\end{pmatrix}+λ\begin{pmatrix}3\\\ 2\\\ 1\end{pmatrix}$$

x = 7 + 3λ
y = 5 + 2λ
z = 3 + λ

Using the equation of the plane
3x+2y+z=6 we get:
3(7+3λ) + 2(5+2λ) + (3+λ) = 6
21 + 9λ + 10 + 4λ + 3 + λ = 6
14λ = -28
λ = -2

$$P = \begin{pmatrix}7\\\ 5\\\ 3\end{pmatrix}-2\begin{pmatrix}3\\\ 2\\\ 1\end{pmatrix}$$

$$P = \begin{pmatrix}7\\\ 5\\\ 3\end{pmatrix}-\begin{pmatrix}6\\\ 4\\\ 2\end{pmatrix} = \begin{pmatrix}1\\\ 1\\\ 1\end{pmatrix}$$

$$P = (1,1,1)$$

Find the distance

$$\vec{|AP|}$$

$$\vec{AP} = P-A = \begin{pmatrix}1\\\ 1\\\ 1\end{pmatrix}-\begin{pmatrix}7\\\ 5\\\ 3\end{pmatrix} = \begin{pmatrix}-6\\\ -4\\\ -2\end{pmatrix}$$

$$\vec{|AP|} = \sqrt{6^2+4^2+2^2} = \sqrt{56}$$

#### Finding angles between 3D Vectors

Find the angle between the vectors p and q in degrees to 1dp where:

$$\underline{p} = 3\underline{i}-2\underline{j}+7\underline{k}$$

$$\underline{q} = \underline{i}+6\underline{j}-4\underline{k}$$

Using knowledge from earlier in Unit 1 we know the equation to find the angle of a vector is:

$$cosθ = \frac{a·b}{|ab|}$$

$$a·b = (3×1)+([-2]×6)+(7×[-4]) = -37$$

$$|ab| = |a|×|b| = \sqrt{3^2+2^2+7^2}×\sqrt{1^2+6^2+4^2} = \sqrt{62}×\sqrt{53}$$

Inputting this into our equation we get:

$$cosθ = \frac{-37}{\sqrt{62}×\sqrt{53}}$$

Rearranging this becomes:

$$θ = cos^{-1}(\frac{-37}{\sqrt{62}×\sqrt{53}}) = 130.2°$$

Position vectors A and B are represented by the vectors a and b. Points C and D have position vectors 4a-b and -10a+5b
Point E lies on the line CD such that the ratio:
CE:ED=1:3

First we should represent this information in a sketch diagram as we do not know the values of A and B but we do know that the Origin (O),C and D form a triangle and that on the line CD lies E in a 1:3 ratio between the two
Using this information we can sketch the graph below to help visualise the vectors

(Remember the ratio 1:3 leaves the fractions 1/4 and 3/4 as there are 4 parts the the ratio[1+3])

Find the expression for the position vector of point E in terms of a and b

$$\vec{|OE|} = \vec{|OC|} + \frac{1}{4}\vec{|CD|}$$

We can do this as on the graph we can see that by tracing the line OC and 1/4 of CD we will land at point E
Filling in the values we know we will find that:

$$\vec{|CD|} = D-C = (-10\underline{a}+5\underline{b})-(4\underline{a}-\underline{b}) = -14\underline{a}+6\underline{b}$$

$$\vec{|OE|} = 4\underline{a}-\underline{b} + \frac{1}{4}(-14\underline{a}+6\underline{b}) = 4\underline{a}-\underline{b} - \frac{7}{2}\underline{a}+\frac{3}{2}\underline{b})$$

$$\vec{|OE|} = \frac{1}{2}\underline{a}+\frac{1}{2}\underline{b}$$

This means that point E lies at:

$$E = \frac{1}{2}\underline{a}+\frac{1}{2}\underline{b}$$

format: plain


text:
Publication
Title
Measurement of the pseudorapidity and centrality dependence of $dE/d\eta$ in PbPb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV extending into the very forward phase-space
Author
Institution/Organisation
CMS Collaboration
Abstract
 The energy density, dE/dh, in PbPb collisions at 2.76 A TeV nucleus-nucleus centerof- mass energy is measured as a function of pseudorapidity, h, and collision centrality using the CMS detector at the LHC. The very large pseudorapidity coverage of this measurement is used as a powerful constraint for various hadronic interaction models. The CASTOR calorimeter extends the h coverage of CMS up to jhj = 6.6, which is only 1.4 units away from the beam rapidity for psNN = 2.76 TeV. This is comparable to the most forward measurements at RHIC. A comparison of the centrality dependence of the very forward energy density to that at smaller pseudorapidities is presented and compared to event generator predictions of models for nuclear collisions. The very wide pseudorapidity range of the presented measurement is unique at this center-of-mass energy and the average energy weighted pseudorapidity can be estimated, which is a useful observable to discriminate between model predictions.
Language
English
Source (series)
CMS physics analysis summary ; CMS PAS HIN-12-006
Publication
S.l. : 2012
Volume/pages
21 p.
UAntwerpen
 Faculty/Department Research group [E?say:metaLocaldata.cgzprojectinf] Publication type Subject Affiliation Publications with a UAntwerp address


text:
# How to Calculate Probability?

Calculating probabilities is perhaps the most common and intuitive application of distributions, probability starts with logic, the formula is p=n/N.
How to Calculate Probability
Probability represents the chance that a possible, but not guaranteed event will occur. For example, you can use probability to help predict what the chances of winning are in such games as dice and poker, or even in larger games, such as the lottery.... More »
Difficulty: Moderately Easy
Source: www.ehow.com
To calculate probability, you want to find the possible results of a specific outcome. The first thing you want to do is figure up all possible variables. Once you know the variables, you will need to figure out the possible outcomes. You can find more information here: http://www.life123.com/parenting/education/probability-statistics/calculate-probability.shtml
Q&A Related to "How to Calculate Probability"
 Calculating probability is a simple calculation represented in most cases by a fraction. All you are doing is finding the number of possible outcomes for a specific case and dividing http://answers.ask.com/Science/Mathematics/how_to_...
 1. Figure out how many outcomes are possible. For example, if you are rolling a die, there are six different possible outcomes (one through six dots) 2. Calculate how many possible http://www.ehow.com/how_8113087_calculate-probabil...
 1 Define your events and outcomes. Probability is the likelihood of one or more events happening divided by the number of possible outcomes. So, let's say you're trying to find the http://www.wikihow.com/Calculate-Probability


text:
Tax Brackets

Tax brackets are remarkably widely misunderstood. I've had numerous conversational partners state that, “I don't want a raise, 'cause that'll put me in the next tax bracket and I'll actually lose money!”

This is profoundly incorrect. Increasing your taxable W-2 income will always increase your post-tax earnings, regardless of your current or future bracket.

As you make more money, of course, your taxes will increase. However, those tax rates are marginal, which means that they only apply to the portion of your income that falls within their bracket.

This table illustrates the tax brackets for a single person filing in 2017:

BracketRate
$0–$9,32510%
$9,326–$37,95015%
$37,951–$91,90025%
$91,901–$191,65028%
$191,651–$416,70033%
$416,701–$418,40035%
$418,401+39.6% Let's work an example by initially supposing that your W-2 income this year will be$90,000 (after any deductions). That puts you in the 25% bracket, so your rates would be:

• 10% on the first $9,325, • 15% on the next$28,625, and
• 25% on the remaining $52,050. If we do some arithmetic: Your tax-home pay, then, comes to$71,761.25, with a total effective tax rate of just under 20.3%.

Now let's suppose that instead you'd earned $100,000, putting you in the 28% tax bracket. Your rates would now be: • 10% on the first$9,325,
• 15% on the next $28,625, • 25% on the next$53,950, and
• 28% on the remaining $8,100. Mathin' it up: That leaves you with a take-home of$79,018.25 at an effective rate of about 21.0%.

Notice that the marginal \$8,100 was taxed at your bracket's 28% rate, but that the rest of your income was taxed at lower rates.

Go ahead and accept that raise! It'll always mean a larger post-tax payment.


text:
Pstat 127 Sec3.pdf - Section 3 Changwei Xu Section Outline I I I Deletion residuals Cooks distance Linear regression with GLM Logistic Regression R

# Pstat 127 Sec3.pdf - Section 3 Changwei Xu Section Outline...

• 28
• 100% (2) 2 out of 2 people found this document helpful

This preview shows page 1 - 8 out of 28 pages.

Section 3 Changwei Xu 1/31/2018
Section Outline I Deletion residuals & Cook’s distance I Linear regression with GLM I Logistic Regression R Example
Deletion residuals & Cook’s distance I We will review deletion residuals and cook’s distance covered in last section. I A concrete example will be discussed.
Case Deletion in Linear Regression Using the notation from the last section, a subscript (i) means “with the ith case deleted,” for examples: I ˆ β ( i ) is the estimate of β computed without case i I X ( i ) is the ( n - 1 ) × p matrix obtained from X by deleting the i th row I Y ( i ) is the ( n - 1 ) × 1 column vector obtained from Y by deleting the i th element In particular, then ˆ β ( i ) = X T ( i ) X ( i ) - 1 X T ( i ) Y ( i )
Deleted Residual If we let I y i denote the observed response for the i th case, and I ˆ y j ( i ) denote the predicted response for the j th case based on the estimated model with the i th case deleted then the i th deleted residual is defined as: d i = y i - ˆ y i ( i )
Studentized Residual I Deleted residuals depend on the units of measurement just as the ordinary residuals do. We can solve this problem though by dividing each deleted residual by an estimate of its standard deviation. That’s where “studentized residuals” come into play. I The studentized residual is defined as t i = y i - ˆ y i ( i ) ˆ σ ( i ) 1 + x T i X T ( i ) X ( i ) - 1 x i where x T i (dimension 1 × p ) is the i th row of X matrix (dimension n × p ) I *A statistic divided by its estimated standard deviation is usually called a studentized statistic , in honor of W.S.Gosset, who first wrote about the t-distribution using the pseudonym Student.


text:
# The velocity of transverse waves on a string

1. A violin string is 35 cm long between its fixed ends and has a mass of 2.0g. The string sounds the note A4 (440 Hz) when played.

## Homework Equations

v=$\sqrt{}F/ρ$

## The Attempt at a Solution

I calculated ρ, mass per unit length, to be .0057 kg/m but do not know how to find the force that I need to plug into the equation.

Another formula (related to the one you mentioned, but derived from the wavelength) gives the wave velocity directly in terms of the freqency and string length. Try this one?

$$f = \frac{v}{2 L}$$


text:
# DokuWiki

It's better when it's simple

Corporate Use

Our Community

plugin:revealjs

# revealjs Plugin

Compatible with DokuWiki

• 2018-04-22 "Greebo" unknown
• 2017-02-19 "<NAME>" yes
• 2016-06-26 "E<NAME>" yes
• 2015-08-10 "Detritus" yes

Converts a DokuWiki page into a Reveal.js slideshow

Last updated on
2017-02-05
Provides
Syntax, Render
Repository
Source
Conflicts with
mathjax

This plugin creates reveal.js presentations from DokuWiki pages. Many reveal.js features such as themes and PDF export are also supported.

The documentation is maintained on Github: DokuWiki plugin reveal.js documentation.

## Installation

Search and install the plugin using the Extension Manager. Refer to Plugins on how to install plugins manually.

#### Change Log

• 2017-02-05
• Major update with all the work done by <NAME> (ottmar [dot] gobrecht [at] gmail [dot] com, https://github.com/ogobrecht)
• Improved slide handling: Since slides are indicated by headers you are relative inflexible regarding the header levels and sizes on vertical (nested) slides - they are normally simply to small
• New conf option horizontal_slide_level: Headers on this level or above starting a horizontal slide. Levels below starting a vertical (nested) slide - no effect on slides, which are indicated by alternative slide indicators
• New conf option enlarge_vertical_slide_headers: Increase headers on slides below horizontal_slide_level - no effect on slides, which are indicated by alternative slide indicators
• New alternative slide indicator:
• ----> opens a new slide with default transition in default speed (open previous slides will be closed implicitly)
• Full example - parameters are parsed dynamically like in CSS, the parameter order is not important: ---- orange wiki:dokuwiki-128.png 10% repeat bg-slide zoom-in fade-out slow no-footer ---->
• All possible HTML color names and codes are supported: red, #f00, #ff0000, rgb(255,0,0), rgba(255,0,0,0.5), hsl(0,100%,50%), hsla(0,100%,50%,0.5)
• Background images are recognized case insensitive by the endings gif, png, jpg, jpeg, svg
• Background image size is recognized by postfix % and px or by keywords auto, contain and cover (cover is the default in Reveal.js) - example: 10% or 250px
• Background image position is recognized by keywords top, bottom, left, right, center (center is the default in Reveal.js) - examples: top left, bottom center
• Background image repeat is recognized by the keyword repeat (no-repeat is the default in Reveal.js)
• Background transition: prefix bg- followed by none, fade, slide, convex, concave or zoom
• Slide transition: none, fade, slide, convex, concave or zoom followed by optional postfix -in or -out for different transitions on one slide
• Transition speed: default, fast, slow
• ---->> opens a new slide container for vertical (nested) slides and a new slide with the given options - example: ---- red zoom ---->>
• The next ---->> will close the previous container (and slide) implicitly
• Technical details:
• In the rendering the slide mode changes from “headers driven” to “special horizontal rule driven” - headers are no longer interesting in this mode for slide changes
• You can create of course a whole presentation with this alternative slide indicator
• if you want to leave this slide mode you need a way to explicit close a slide or container:
• <<---- closes a slide container (and possibly open slide inside)
• <---- closes a slide
• Improved existing background indicator: {{background>orange bg-zoom}}
• allowing now all options from the new slide indicator (see above)
• Improved wiki page rendering:
• New conf option show_slide_details (default on): Show slide details on wiki page: start of slide and options, background preview, start of notes
• Improved section editing:
• Slide background definition or alternative slide indicators must be noted before the headers/slide content
• The normal section editing has the problem, that it is starting on the headers, so the background definition or alternative slide indicators are on the wrong section
• With the section editing improvement the whole slide can be edited at once including the background definitions
• New conf option for the start button:
• You can now select between the default one(start_button.png), the one from the s5 plugin (start_button.screen.png) and an own one (start_button.local.png), which is upgrade safe and must be copied into the plugin directory
• If you are able to edit the page, then a export to PDF link is rendered under the start slideshow button - this is the reveal.js default export and works only in Chromium and Chrome - more infos here
• Introducing new syntax for speaker notes:
• <notes> - no parameters
• No changes on wiki pages
• On a slideshow the content is wrapped into <aside class="notes">
• Lists in notes are always NOT incremental, because the list is unvisible and you would have to press the next key for each entry without any obvious effect
• Introducing new syntax for fragments:
• <fragment> for inline usage (only formatting and substitutions supported)
• <fragment-block> for any wiki content
• <fragment-list> to overwrite the global option build_all_lists (if false)
• <no-fragment-list> to overwrite the global option build_all_lists (if true)
• Support for style and index where possible - see also the source code file example_presentation.dokuwiki and this reveal online demo
• Improved blockquote handling:
• The nesting is suppressed on the slideshow to support the way Reveal is showing notes
• Improved DokuWiki text formatting on the slides:
• Underlined text is now underlined and no longer italic
• Subscript and superscript are now a bit smaller than the default text
• New conf option for image borders:
• In DokuWiki in the default theme images have no borders
• In Reveal the images have borders
• Now you can configure this globally in the plugin settings and since all options are overwritable per wiki page you can have different settings on different pages/presentations
• New conf options for slide size:
• Default size in Reveal.js is 960×700
• Sometimes you need to align this if you have content that does not fit on this size
• As all options this one is overwritable per page/presentation - see also example_presentation.dokuwiki
• New theme dokuwiki:
• This is a copy of the theme solarized: the only change is the aligned background color, which match the background color of the default DokuWiki theme
• Images are looking good with activated borders
• 2016-09-15
• Better IE support
• Add syntax to set and override all options for a presentation
• 2016-09-07
• Add option to open presentation in new window and make this behavior default
• Nicer description of the reveal.js options
• 2016-02-11
• fix small method signature inconsistency
• 2016-02-04
• 2015-07-09
• 2015-03-28
• add option to insert a button which starts the presentation together with theme selection similar to the S5 Presentations Plugin
• slide background are also supported
• 2015-01-27
• update to reveal.js 3.0
• new default theme “white”
• more configuration options
• 2014-12-09
• Add configuration option to build all lists by default
• Remove black border in theme “beige_white”
• 2014-10-31
• Load MathJax over secure connection if available
• 2014-10-28
• Add support for the Reveal.js (highlight.js) syntax highlighter
• 2014-10-11
• Initial release


text:
# Department of Mathematics

## Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

### DOI

We provide lower bounds for the sum of the negative eigenvalues of the operator | σ· pA| 2 s- Cs/ | x| 2 s+ V in three dimensions, where s∈ (0 , 1] , covering the interesting physical cases s = 1 and s = 1/2. Here σ is the vector of Pauli matrices, pA= p- A, with p= - i∇ the three-dimensional momentum operator and A a given magnetic vector potential, and Cs is the critical Hardy constant, that is, the optimal constant in the Hardy inequality | p| 2 s≥ Cs/ | x| 2 s. If spin is neglected, results of this type are known in the literature as Hardy–Lieb–Thirring inequalities, which bound the sum of negative eigenvalues from below by -Ms∫V-1+3/(2s), for a positive constant Ms. The inclusion of magnetic fields in this case follows from the non-magnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for 1 / 2 ≤ s≤ 1 we are able to express the bound purely in terms of the magnetic field energy ‖B‖22 and integrals of powers of the negative part of V.

Original language English Communications in Mathematical Physics 365 2 651-683 33 0010-3616 https://doi.org/10.1007/s00220-018-3204-y Published - 2019

### Research areas

• ASYMPTOTICS, BOUND-STATES, COULOMB-SYSTEMS, FERMIONS, HEAVY-ATOMS, INSTABILITY, NONHOMOGENEOUS MAGNETIC-FIELDS, RELATIVISTIC MATTER, SIMPLE PROOF, STABILITY

Citationformats

ID: 131295553


text:
# All Questions

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### Do companies like CipherCloud really have the option to use homomorphic encryption?

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### Fractal Merkle Traversal

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### Pre-image resistant but not 2nd pre-image resistant?

Are there any cryptographic hash functions for which there is a known pre-image attack, or a known second pre-image attack, but not both? The attack doesn't have to be practical - just anything that ...
236 views

### How to use salt when there is only one user

I am encrypting files for storage in an untrusted location, using a custom Java program to do that. There is only one user, but there are many files. I am using AES in CBC mode with PKCS5 padding, and ...
2k views

### RC2, RC4, RC5 key length

While reading the Wiki page on RC4 I noticed that the key size must be in the range of 40–2,048 bits. So should I understand like this, that I can have a key that has length of, lets say 333 bits? ...
424 views

### How does certificate-based encryption solve key revocation problem?

How does certificate-based encryption solve key revocation problem? For example, if a user accidentally reveals its secret key or an attacker compromises it, the user may request revocation of its ...
1k views

### Why does DES use exactly 16 rounds?

Why does DES use exactly 16 rounds and not more or less than 16? What is the significance of using 16 rounds?
135 views

### Verify Messages to Embedded Device

I'm building an embedded device that I plan on distributing. Periodically the device will poll my server to check for updates and commands. I'd like the device to verify that any messages(JSON strings ...
7k views

### How does DES decryption work? Is it the same as encryption or the reverse? [duplicate]

If DES decryption is the same as encryption done in reverse order, then how can the reversed S-Box convert 4 bits into 6 bits?
351 views

### How Were secp*k1 elliptic curve generators chosen?

The Koblitz elliptic curves specified in the SEC2 document https://docs.google.com/viewer?url=http%3A%2F%2Fwww.secg.org%2Fcollateral%2Fsec2_final.pdf all have the nice feature that the parameters are ...
736 views

### Can AES decryption be used as encryption?

Definition E: AES encryption D: AES decryption x: plain text y: encrypted text k: key In original AES cipher, encryption: y = E(x, k) decryption: x = D(y, k) Then I define the "reverse AES ...
187 views

### What are the security effects of reusing a public RSA key to encrypt large data by blocks? [duplicate]

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215 views

### Does impersonating an SRP server give you enough information for an off-line dictionary attack?

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628 views

### How to supply the GCM authentication tag to the OpenSSL command line tools?

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272 views

### why AKS is so slow in practice? [closed]

This note (http://maths-people.anu.edu.au/~brent/pd/primality4.pdf) states that AKS is not practical. However, it is known that AKS runs in polynomial-time, and I cannot understand where the slowness ...
151 views

### Can an encrypted nonce be used for authentication?

Would a scheme like the following work to allow authentication of a sender: A nonce is appended to data, and the result is encrypted. The encrypted pack is sent to the other party. The sender then ...
216 views

### Is this method of deterministically using CBC secure?

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### Alice's forgetful banking

Alice has a bank account number, but has forgotten which bank it is for. There are 4 banks, run by Bob, Carlos, David, and Eve. She could find out by going to all of the banks and asking if they have ...
146 views

### Could using sequential plaintext uncover AES key?

I am encrypting a string in a format like this: ...
2k views

### Learning to encrypt/encryption algorithms [closed]

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144 views

### Can OTP still be broken if there are several messages being sent with only a slight variation in a password?

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text:
# Chaitanya's Random Pages

## October 28, 2015

### The product of distances to a point from vertices of a regular polygon

Filed under: mathematics — ckrao @ 11:12 am

Here is a cool trigonometric identity I recently encountered:

$\displaystyle \prod_{k=1}^n \sin \frac{(2k-1)\pi}{4n} = \prod_{k=1}^n \cos \frac{(2k-1)\pi}{4n} = \frac{\sqrt{2}}{2^n}.$

For example, for $n = 9$:

$\displaystyle \sin 5^{\circ} \sin 15^{\circ} \sin 25^{\circ} \ldots \sin 85^{\circ} = \frac{\sqrt{2}}{2^9}.$

After thinking about it for some time I realised that the terms on the left side can each be seen as half the lengths of chords of a unit circle with 4n evenly spaced points that can then be rearranged to be distances from a point on the unit circle to half of the points of a regular (2n)-gon, as shown in the figure below.

With the insight of this figure we then write

\begin{aligned} \prod_{k=1}^n \sin \frac{(2k-1)\pi}{4n} &= \left(\prod_{k=1}^{2n} \left| \sin \frac{(2k-1)\pi}{4n} \right|\right)^{1/2} \quad \text{ (all terms are positive)}\\ &= \prod_{k=1}^{2n} \left| \frac{\exp(i\frac{(2k-1)\pi}{4n}) - \exp(-i\frac{(2k-1)\pi}{4n})}{2i} \right|^{1/2} \\ &= \frac{1}{2^n} \prod_{k=1}^{2n} \left| \exp\left(-i \frac{(2k+1)\pi}{4n}\right) \left(\exp\left(i\frac{4k\pi}{4n}\right) - \exp\left(i\frac{2\pi}{4n}\right)\right) \right|^{1/2}\\ &= \frac{1}{2^n} \prod_{k=1}^{2n} \left| \exp\left(-i \frac{(2k+1)\pi}{4n}\right) \right|^{1/2} \left| \left(\exp\left(i\frac{4k\pi}{4n}\right) - \exp\left(i\frac{2\pi}{4n}\right)\right) \right|^{1/2}\\ &= \frac{1}{2^n} \prod_{k=1}^{2n} \left| \left(\exp\left(i\frac{2k\pi}{2n}\right) - \exp\left(i\frac{\pi}{2n}\right)\right) \right|^{1/2}\\ &= \frac{1}{2^n} \left|\prod_{k=1}^{2n} \left(z-\exp\left(i\frac{2k\pi}{2n}\right)\right) \right|^{1/2} \quad \text{where }z = \exp\left(i\frac{\pi}{2n}\right)\\ &= \frac{1}{2^n} |(z^{2n}-1)|^{1/2}\\ &= \frac{1}{2^n} |-1-1|^{1/2}\\ &= \frac{\sqrt{2}}{2^n}. \end{aligned}

(The cosine formula can be derived in a similar manner.)

In general, the product of the distances of any point $z$ in the complex plane to the n roots of unity $\omega_n$  is

$\displaystyle \prod_{k=0}^{n-1} |z-\omega_n| = |z^n - 1|.$

The above case was where $z^n = -1$. Two more cases are illustrated below, this time for n = 10. In the left example the product of distances is

$\displaystyle \prod_{k=0}^9 |(1+i)-\exp(2\pi i k/10)| = |(1+i)^{10}-1| = 5\sqrt{41}$

while for the right example it is

$\displaystyle \prod_{k=0}^9 |1/2-\exp(2\pi i k/10)| = |(1/2)^{10}-1| = 1023/1024.$

Note that earlier in the year I posted on the distances to a line from vertices of a regular polygon.


text:
# Code Simplicity

## The Universal Laws of Nature

Software

## Purpose

• I'm @zachfedor, a teacher.
• I know that I know nothing.
• I want to hold a discussion, not tell you what you're doing wrong.
• I like playing the devil's advocate!

## Credit

By <NAME>

## Context

• The author was a Chief Architect of the Bugzilla Project
• There's no code samples, just anecdotes, observations, and his conclusions.
• It's less than 70 pages, so go read the whole thing!

## Labels

• Fact: a true statement, piece of true information
• Rule: advice to guide a specific decision, won't predict the future
• Law: fact that will always be true, prediction of what will happen in the future

Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.

<NAME>, Meditations

# Facts, Rules, and Laws

Controversial

Notable

## Law

The purpose of software is to help people.

## Fact

Every programmer is a designer.

## Rule

Design is not a democracy. Decisions should be made by individuals.

## Rule

The quality of your design should be proportional to the lifespan of the code.

## Rule

You are safest if you don't attempt to predict the future and only act on immediate information.

## Law

The longer your software exists, the more likely it is that any piece will have to change.

## Law

The chance of introducing a bug is proportional to the size of changes you make.

## Law

The desirability of a change is mostly affected by the effort of maintenance.
D = \frac{V_n + V_f}{E_i + E_m}
D = \frac{V_f}{E_m}

## Rule

The best design allows for the most change in the environment with the least change in code.

## Fact

The difference between a bad programmer and a good programmer is understanding.

## Fact

Simplicity is relative.

## Law

The ease of maintenance of software is proportional to the simplicity of its parts.

## The End

• Thanks for sharing your thoughts!
• This was a small sample of the book so go read it if you're curious for more.
• Let's keep the discussion going on the #front-end channel in Slack!

By <NAME>

# Code Simplicity

The Universal Laws of Software adapted from the book by <NAME>

• 526


text:
Uncertainty consistency?

1. The problem statement, all variables and given/known data

I am given a set of x and y values x: (1,2,3,etc.) y: (1.2,2.2,3.1,etc.) with a given uncertainty and am asked

a) find the best fit

b) at what prob can you rule out a 5% higher slope

c) is the stated uncertainty consistent with the data?

I can find the best fit relatively easily by minimizing chi-sqd and setting the derivatives to 0. I am confident in my result as it matches with the graph given by excel.

For part 2 I tried putting a higher value for the given slope into the chi-sqd equation and checking a chart but that didn't give me a reasonable answer (not completely sure about the degree of freedom)

I am not sure how to approach the third part. I have computed the errors in the fitting coefficients but that doesn't seem to play into it.

Thanks,

• Because this is a homework/self-study question, you should add the self-study tag to your question. See stats.stackexchange.com/tags/self-study/info Jul 19 '14 at 17:55
• Sure thing, my mistake
– Jim
Jul 19 '14 at 17:58

Without more context is difficult to be sure. There are many models that could be fit, and many criteria used to fit those models.

Part (b) suggests that the model is linear ("slope") which suggests that the criterion is Least Squares.

Part (c) reinforces the hint that Least Squares is the criterion, because the MS Residual estimates the uncertainty from the data. This could be used for a $\chi^2$ test of the uncertainty (with $n-2$ df).

• Yes a linear model fits the data set, the slope is around 1 and the y-intercept is 0. Not sure what other context to give? I put a value of a 5% higher slope into the chi-sqd equation and got a number like 90, which is untenable.
– Jim
Jul 19 '14 at 22:41
• Why are you using $\chi^2$? I'm at a bit of a loss to figure out how you are supposed to test the hypothesis $\beta > 1.05\, \hatP\{\beta}$ This is a data-dependent hypothesis, and so isn't covered by Neyman-Pearson theory that I am aware of. If you let $\sigma_=o$ be the specified variance (uncertainty), then $\frac{SSRes}{\sigma_o^2} \sim \chi^2_{n-2}$ when $H_o:\sigma = \sigma_o$. Jul 20 '14 at 23:13


text:
923 Pages

## Dr <NAME>

### What is Snell's Law?Edit

Snell's Law is used to find the relationship between the angle of Incidence and Refraction and the indices of refraction of multiple mediums. Snell's Law can be used to find the refraction of a light ray in any situation, no matter what the different mediums(substances the light passes through) are.
$n_1\sin(\theta_1) = n_2\sin(\theta_2)\ ,$
Snells Law() of llama

### Who created Snell's Law? What is it's history?Edit

Snell's law was first discovered by <NAME> in 984. Sahl used it to work out the shapes of Anaclastic Lenses (lenses that focus light with no geometric aberrations). Later on, it was discovered again by <NAME> in 1602, but it was not published. In 1621, it was discovered another time by <NAME>, in a form that was basically written in a similar mathematical format. This work was never published during Snell's life. <NAME> independently derived the law in terms of sines in his 1637 treatise Discourse on Method, and used it to solve a range of optical problems. Despite the many people who have worked on Snell's Law over the years, only Willebrord Snell is mentioned in its name.
<NAME>
<NAME>

Community content is available under CC-BY-SA unless otherwise noted.


text:
# Course:CPSC522/Value of Information and Control

## Value of Information and Control

The values of information and control quantify how access to the value of a variable, or the ability to control it, can improve an agent's utility.

Principal Author: <NAME>
Collaborators:

## Abstract

Information value theory provides a basis for evaluating actions which may provide little or no direct reward, but which provide an agent with information. Since later actions can be improved by conditioning on this information, these information-seeking actions can have indirect value[1], and therefore can be part of an optimal strategy. By quantifying this `value', it is possible for the agent to make comparisons between information-seeking actions and any other actions, and so identify the optimal strategy. Unlike the information theory developed by Shannon, which evaluates informativeness based purely on the probability distributions, information value theory also accounts for how the information will affect later decisions and rewards, which is necessary for determining an optimal strategy. Correspondingly, control value theory quantifies how beneficial it can be to control random variables. This control may be used to directly affect the utility, or to gain more information.

### Builds on

We discuss these concepts in the context of decision networks, which build on Bayesian networks.

### Related Pages

Quantifying the value of information has applications for decision making under uncertainty (e.g. in Partially Observable Markov Decision Processes and reinforcement learning) and for allocating sensors or processing (see bounded rationality).

## Value of Information

Access to information can have a clear value by allowing agents to make better decisions. The value of the information depends on how much better it can make the decisions: the increase in expected utility when decisions are conditioned on the information. By quantifying this value, the agent is able to actively choose to seek information, allowing a higher expected utility. For example, consider betting on a sports match: if you knew the outcome in advance, you could place a bet and be certain to make a profit. However, if you couldn't predict the outcome, the optimal strategy is likely to be not to make a bet at all. The value of this information then corresponds to the most that you would be prepared to pay to a fortune teller/match fixer to tell you the outcome.

We will now consider a more precise definition involving decision networks.

### Definition

A distinction is made between the value of perfect information (also known as the value of clairvoyance), and the value of imperfect information[2]. Consider a non-forgetting decision network, ${\displaystyle N}$, which includes a decision variable, ${\displaystyle D}$, and a random variable, ${\displaystyle X}$. The value of perfect information about ${\displaystyle X}$ for ${\displaystyle D}$ is defined as the difference between:

• the value of the decision network when an arc is added from ${\displaystyle X}$ to ${\displaystyle D}$ (and from ${\displaystyle X}$ to all later decision variables such that ${\displaystyle N}$ remains a non-forgetting network)
• the value of the decision network without an arc from ${\displaystyle X}$ to ${\displaystyle D}$

This leads to an alternative (but equivalent) definition: the value of perfect information of ${\displaystyle X}$ about ${\displaystyle D}$ is the upper bound on the price that a rational decision-maker would be willing to pay to know ${\displaystyle X}$ before making decision ${\displaystyle D}$.

The value of imperfect information for ${\displaystyle X}$ is defined as the value of the information gained from a noisy measurement of ${\displaystyle X}$.

### Characteristics

1. The value of information is always non-negative because, in the worst case, an agent can ignore the information and so the value would be 0. Therefore, acquiring information is never harmful for a rational agent.
2. Additionally, the value of perfect information is always greater than the value of imperfect information. This means that the value of perfect information can be used to provide an upper bound on the value of imperfect information[1]. One could use this to, for example, rule out buying a noisy sensor of ${\displaystyle X}$ (which would provide imperfect information about ${\displaystyle X}$) if it would cost more than the value of perfect information about ${\displaystyle X}$.
3. The value of information about ${\displaystyle X}$is only non-zero if decisions made depend on the value of ${\displaystyle X}$.

### Example

Consider the decision networks in Figures 1-4, based on [1] and explained in detail in decision networks. A decision, Umbrella, must be made, determining whether or not to bring an umbrella. This should be done to maximise utility, which depends on both Weather and Umbrella, as shown in the table below:

Weather Umbrella Utility
no rain take it 20
no rain leave it 100
rain take it 70
rain leave it 0

The joint distribution over the weather and the weather forecast is shown below. It can be seen that the forecast provides some information about the weather, but not perfect information as, for example, Forecast can be rain when Weather is not rain.

Weather Forecast Probability
no rain sunny 0.49
no rain cloudy 0.14
no rain rainy 0.07
rain sunny 0.045
rain cloudy 0.075
rain rainy 0.18

#### Value with no Information

Figure 1. A decision, Umbrella, must be made without any information. The utility depends on both Umbrella and Weather, the true weather. Based on Section 9.3.3 of "Artificial Intelligence: Foundations of Computational Agents"

In the Figure 1, the decision must be made without knowledge about Weather or Forecast. The value for each possible decision (the expectation of the utility) is shown below:

Umbrella Value
leave it 70
take it 35

So the optimal decision is to leave the umbrella, leading to an expected utility of 70.

#### Value with Observed Forecast

Figure 2. A decision, Umbrella, must be made conditioned on a weather forecast, Forecast. The utility depends on both Umbrella and Weather, the true weather. Based on Section 9.3.3 of "Artificial Intelligence: Foundations of Computational Agents"

In Figure 2, the decision can depend on Forecast. As shown in , this gives the following decisions are optimal for each value of Forecast:

Forecast Umbrella Value
sunny leave it 91.6
cloudy leave it 65.1
rainy take it 56

Taking an expectation over the values of Forecast, the expected utility is 77. This is 7 greater than the value without conditioning on the Weather, and so the information value of Forecast for Umbrella is 7. Note that, since Forecast is essentially a noisy measurement of Weather, this can be viewed as the value of perfect information about Forecast or, alternatively, as the value of imperfect information about Weather. Also, note that it is positive, showing that information about the weather forecast improves decision-making about bringing an umbrella (and satisfying the requirement that the value of information is non-negative).

#### Value with Observed Weather

Figure 3. A decision, Umbrella, must be made conditioned on perfect information about the weather. The utility depends on both Umbrella and Weather, the true weather. Based on Section 9.3.3 of "Artificial Intelligence: Foundations of Computational Agents"

Now consider Figure 3. In this, the agent is able to condition its decision on the true value of Weather, rather than relying on the imperfect information contained by Forecast. This leads to the decisions shown below:

Weather Umbrella Value
no rain leave it 100
rain take it 70

Taking an expectation over Weather reveals that the value is 91. This is 21 greater than without information, and so the value of perfect information about Weather is 21. This is greater than the value of the imperfect information about Weather we considered previously (ie the perfect information about Forecast), satisfying the characteristic that the value of perfect information is an upper bound on the value of imperfect information.

#### Value with Observed Weather and Forecast

Figure 4. A decision, Umbrella, must be made conditioned on perfect information about both the weather and the weather forecast. The utility depends on Umbrella and Weather, the true weather. Based on Section 9.3.3 of "Artificial Intelligence: Foundations of Computational Agents"

Now consider Figure 4, in which the agent has access to both Weather and Forecast when making the decision. This leads to the decisions and values shown below:

Weather Forecast Umbrella Value
no rain sunny leave it 100
no rain cloudy leave it 100
no rain rainy leave it 100
rain sunny take it 70
rain cloudy take it 70
rain rainy take it 70

Taking an expectation over Weather and Forecast gives a value of 91. This is identical to the situation where we observed only Weather. This can be explained by the fact that, in this case, we already had perfect information about Weather. Since Forecast is simply a noisy measurement of Weather, and does not otherwise affect the utility, information about it is redundant when we already know Weather. Therefore, we can say that the value of information about Forecast is zero when there is an arc from Weather to Umbrella. Note that, as with any random variable with a value of zero, the decisions made do not depend on the value of Forecast (see the table above).

### Sequences of Decisions

The example above dealt with a decision network with a single decision variable. The value of information theory can also be applied when decisions are made sequentially. In this case, there is the added complication that the time when we obtain the information becomes important. For example, it is important to have information about the weather before deciding whether or not to take an umbrella, even if more decisions will be made later. Therefore, the value of information can be different depending on when it is obtained.

## Value of Control

Figure 5. Decision network with Forecast turned into a decision variable. It still has the same parent as before. An additional arc has been drawn from the parent, Weather, to Umbrella, to ensure that the network remains non-forgetting. Based on Section 9.3.3 of "Artificial Intelligence: Foundations of Computational Agents"

Similarly to the value of information quantifying the value of knowledge about a random variable, the value of control quantifies the value in being able to control it. In the context of decision networks, the value of control is defined as the increase in value when a random variable is replaced with a decision node and edges are added to make it a no-forgetting network (so that later decision variables have access to all random variables that previous decision variables had access to - see the page on decision networks). The properties of the new network depend on which nodes are used as parents to the new decision variable. We now discuss two possible cases:

### Control with Parents Maintained

This means that the decision can be made using all the information that the random variable could depend on. In the context of decision networks, the decision variable has the same parents as the random variable it replaces. In this case, controlling the variable may also involve adding edges from the parents to later decision variables to maintain the no-forgetting property.

Figure 6. Decision network with Forecast turned into a decision variable. Forecast now has no parent nodes. Based on Section 9.3.3 of "Artificial Intelligence: Foundations of Computational Agents"

When the parents are maintained, the value of control is always non-negative, and no less than the value of perfect information about the same node.

#### Example

Consider the decision network in Figure 4. It is based on the example from the page on decision networks but Forecast has been turned into a decision node. Its parent nodes are the same as before, and an additional arc has been drawn from Weather to Umbrella to ensure that this is a no-forgetting network. This additional arc ensures that the decision node, Umbrella, has perfect information about the weather, and so the value is 91 - meaning that the value of control in this context is 14. As with any case where the parents are maintained, the value of control of Forecast (14) proves to be no less than the value of perfect information (0, since Forecast was already a parent of Umbrella). Note that, although, the value of control of Forecast is equal to the value of perfect information about Weather in this case, there is not always such a correspondence. For example, if Weather was controlled, it could always be set to not rain and the value of control would be 30, greater than the value of perfect information about any node.

### Control with Different Parents

If the decision variable does not have all of the same parents as the random variable, the value of control can be less than the value of information and can, in fact, be negative.

#### Example

As an example, consider the decision network in Figure 5, where Forecast has been turned into a decision variable with no parents. Forecast is now controlled, but cannot depend on Weather. Therefore, Umbrella cannot depend on Weather either and so the value is 70. Since the value of the decision network was 77 before Weather was controlled, the value of control of Weather is -7. This is negative because Umbrella has gone from having imperfect information about Weather to having no information, whilst controlling Forecast cannot increase the utility on its own. The fact that the value of control is negative illustrates that control is not necessarily helpful if the parents change, and shows the importance of specifying the parents of controlled variables when evaluating the value of control.

## Annotated Bibliography

1. <NAME> & <NAME>, "Artificial Intelligence: Foundations of Computational Agents", Chapter 9.4
2. *<NAME>. Information value theory. IEEE Transactions on systems science and cybernetics. 1966 Aug;2(1):22-6.


text:
# Class 10: Logistic Regression and One-sample tests

### Learning Objectives

• Apply logistic regression to model the probability of an event as the response variable, both with and without nusiance variables.
• Understand the log-odds of an event and how it is used in logistic regression.
• Apply one-sample tests for a mean.

### Modeling probabilities

Looking back through your notes, you’ll notice that we have not yet worked with an inference problem where the response (DV) variable is categorical and the effect (IV) variable is continuous. To do this, we will make use of logistic regression.

If you are only concerned with inference, logistic regression is very straightforward and virtually unchanged from linear regression. Let’s take the cars dataset one last time and look at whether a car has been awarded a “Green Energy Award” as a function of its fuel efficency in the city:

The p-value here tells us the probability of observing an effect at least as strong as shown in the data if there is no change in the probability of a green award as a function of the city fuel efficency.

We can also extend this to a categorical variable (and it becomes an F-test):

And we can include any number of nusiance variables into the model:

So far this is fairly straightforward, and most of the time this is the most import part (inference) of logistic regression in applications.

## Logistic regression

How does logistic regression actually work and what are the point estimates telling us? Let’s look at these two questions now.

Assume for a moment that we have a variable p that tells us the probability of observing a particular event. The odds of this event is the ratio of p divided by 1-p (the probability of not observing p):

This quantity is often used in gambling contexts. Notice that an odds of 1 indicates that the probability is equal to 0.5; it is just as likely to observe the event as it is to not observe the event.

The log odds is simply the (natural) logarithm of the odds:

Logistic regression tries to model the logs odds of an event. Specifically, it tries to build a model that look like:

For some variable X. Why model the log odds and not the probability itself? The probability can only be between 0 and 1, whereas the log odds can range over and real number. The right-hand side above could take on any value (depending on the inputs X), and so it is much more mathematically sound to model the log odds rather than the probability directly. The formulation given here also a high-degree of symmetry about what class we pick as the event (specifically, if we flip 0 and 1 in the input, the values of the slopes and intercepts simply switch signs).

Looking again at the logistic regression of the green award as a function of the city variable, how do we interpret the parameter value?

We say that the logs odds increases by 0.3892 for every unit of increase in the city miles per gallon fuel efficency of the car. Is this very satisfying? Probably not, because there is not way to directly map this statement into things about probabilites. It is just what we are stuck with and the best that we can do.

## One-sample tests

There is one final type of test that we have not seen yet that is fairly common in applied statistics. It is actually one of the easiest tests to understand and is often the first type of test introduced in a statistics course. I choose to do it last because while the theoretical concepts are easy to understand I find it harder to explain why the test might be useful.

Specifically, consider the hypothesis test for a variable x defined by:

For some value of the parameter mu. This at first looks a lot like the tests we have already done, but it involves testing the exact value of mean variable’s mean rather than looking at the difference between the means of two variables. This is called a one-sample test.

The two most common one-sample tests are: the one-sample t-test (parametric) and the Wilcoxon test (non-parametric). To run this with the tmodels package, use the function tmod_t_test_one and just put the number 1 on the right-hand-side. There are no independent variables in this test; the 1 serves as just a place-holder.

The point estimate gives the mean of the city variable and the confidence interval gives a range of guesses for the mean (NB: In my experience, the confidence intervals are actually more common and useful with one-sample tests than the p-value). We can change the value of mu that defines the null hypothesis by inputing the parameter h0 in R:

The non-parametric test works similarly: