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### Home > CC2 > Chapter Ch2 > Lesson 2.2.5 > Problem2-92

2-92.

Simplify each expression.

1. $-32+|10+2|$

Treat the absolute value signs like parentheses.
Complete the operations inside first, and then make the simplified term positive.

$-32+\mid10+2\mid=-32+\mid12\mid=-32+12$

Then add the two terms.

$-20$

1. $17\frac{1}{3}+\left(-50\frac{1}{3}\right)-5\frac{1}{2}$

Simplify this one as you would any expression like it.

Add the first two terms together first to eliminate the fractions. Then subtract the third term.

$-38\frac{1}{2}$

1. $-5.37+8.14-1.89$

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## Tags » Sequences And Series

#### Sum of the Reciprocals of the Central Binomial Coefficients

In this post we prove the formula for the sum of the reciprocals of the central binomial coefficients :

(Of course, the sum of the central binomial coefficients themselves does not converge.) 108 more words

Binomial Coefficients

#### A Proof of Dobinski's Formula for Bell Numbers

Dobinski’s formula entails the following infinite series expression for the nth Bell number :

In this post we’ll work through a proof of Dobinski’s formula.   97 more words

Sequences And Series

#### calculator and arithmetic sequences

The 26th term of an arithmetic sequence is 85, and the 16th term is 55.

Find the first term and the common difference.

• Name u1 the first term and d the common difference…
• 55 more words
Calculator

#### An Explicit Solution to the Fibonacci Recurrence

The Fibonacci sequence is a famous sequence of numbers that starts 1, 1, 2, 3, 5, 8, 13, 21, and continues forever.  Each number in the sequence is the sum of the two previous numbers in the sequence.   711 more words

Sequences And Series

#### Calculating SECA Offsets for Clergy

Recently my pastor posed a problem to me: How to calculate his SECA offset?  The answer can be found using algebra or using infinite series.  (For the non-math inclined, skip to the bottom of this post for an answer.) 642 more words

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#### February 2016

As I hope you’ve noticed there is a new pull-down on the navigation bar called “Website.” For some years I’ve had a website at linmcmullin.net that lately I’ve been neglecting. 203 more words

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?

Free Version
Moderate

# Perfect Competition and Monopoly: Profit-Maximization for Firms

APMICR-GO40RI

Which of the following is true at the profit-maximizing output for a perfectly competitive firm but NOT for a monopoly, if both are earning economic profits?

A

$MR = MC$

B

$P > ATC$

C

$P = MC$

D

$P >MC$

E

$P = ATC$


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# Revision history [back]

Actually, some computations can be done by sympy, see for example:

sage: integral?

However sympy is not imported in Sage by default, i guess the main reason is startup speed. If you want to use sympy within Sage, just do:

sage: import sympy

Now, if you have an element of the Symbolic Ring, you can transform it into a sympy object as follows:

sage: a = cos(x) + pi
sage: b = a._sympy_()
sage: b
cos(x) + pi

You can check:

sage: type(b)

You can come back to the symboloc ring as follows:

sage: b._sage_()
pi + cos(x)

Actually, Concerning relations between Sage, maxima and ginac, you can have a look to this question.

Concerning sympy, some computations can be done by sympy, but you need to specify it, see for example:

sage: integral?

However sympy is not imported in Sage by default, i guess the main reason is startup speed. If you want to use sympy within Sage, just do:

sage: import sympy

Now, if you have an element of the Symbolic Ring, you can transform it into a sympy object as follows:

sage: a = cos(x) + pi
sage: b = a._sympy_()
sage: b
cos(x) + pi

You can check:

sage: type(b)

You can come back to the symboloc symbolic ring as follows:

sage: b._sage_()
pi + cos(x)


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# multilayer

## Aplicaciones ópticas de láminas delgadas

En los últimos años se ha conseguido un gran avance en la tecnología de capas delgadas para aplicaciones ópticas. Esto es debido a que las aplicaciones tecnológicas han ido aumentado el nivel de exigencia en lo que se refiere a la calidad y …

## Optical and structural properties of Sb$_2$S$_3$/MgF$_2$ multilayers for laser applications

Multilayers of MgF2 and Sb2S3 have been obtained by physical vapour deposition on glass substrates. Changes in the optical and structural properties have been studied as a function of annealing temperature and the number of layers. A drastic …

En los últimos años se ha conseguido un gran avance en la tecnología de capas delgadas para aplicaciones ópticas. Esto es debido a que las aplicaciones tecnológicas han ido aumentado el nivel de exigencia en lo que se refiere a la calidad y …

## Electric Field Enhancement of the Rabi Splitting in a Superlattice-Microcavity System

We have observed a large coupling between the photonic mode of a microcavity and Stark-localized excitons from a superlattice embedded in it. The coupling is significantly stronger than in comparable microcavities that contain quantum wells solely at …

## Depth-Resolved Microspectroscopy of Porous Silicon Multilayers

We have measured micro-photoluminescence (PL) and micro-Raman spectra on the cross section of porous silicon multilayers to sample different layer depths. We find noticeable differences in the spectra of layers with different porosity, as expected …

## Depth-Resolved Microspectroscopy of Porous Silicon Multilayers

*Appl. Phys. Lett.* **75**, 977--979 (1999)

## Wannier-Stark localization and Bloch oscillations

This chapter discusses the effects of an electric field on the electronic and optical properties of semiconductor superlattices. Here an ideal superlattice is considered to be an infinite periodic series of strongly coupled quantum wells. The …

## Miniband formation at finite electric fields in a graded-gap superlattice

We have investigated a graded-gap superlattice with differential photocurrent spectroscopy at 77 K. The lowest conduction subband exhibits with increasing electric field a transition from localized to extended miniband states and back to localized …

## Local origin of photocurrent in semiconductor superlattices

To trace the origin of photocurrent (PC) in semiconductor superlatices, we have studied the PC spectra as a function of the applied electric field in a novel compositionally graded superlattice, where each well produces a separate peak. The sample …


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# How to find $x^4+y^4+z^4$ from equation?

Please help me. There are equations: $x+y+z=3, x^2+y^2+z^2=5$ and $x^3+y^3+z^3=7$. The question: what is the result of $x^4+y^4+z^4$?

Ive tried to merge the equation and result in desperado. :( Please explain with simple math as I'm only a junior high school student. Thx a lot

solution 2: since $$x^4+y^4+z^4=(x^2+y^2+z^2)^2-2x^2y^2-2y^2z^2-2x^2z^2$$ and $$(xy+yz+xz)^2=x^2y^2+x^2z^2+y^2z^2+2xyz(x+y+z)$$ since $$xy+yz+xz=2,xyz=-\dfrac{2}{3}$$ so $$x^2y^2+y^2z^2+x^2z^2=8$$ so $$x^4+y^4+z^4=5^2-2\cdot 8=9$$
since $$x^2+y^2+z^2=(x+y+z)^2-2xy-2yz-2xz\Longrightarrow xy+yz+xz=2$$ $$x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-xz)=(x+y+z)^3-3(xy+yz+xz)(x+y+z)$$ so $$7-3xyz=27-18\Longrightarrow xyz=-\dfrac{2}{3}$$ use $$x^4+y^4+z^4=(x+y+z)(x^3+y^3+z^3)-(xy+yz+xz)(x^2+y^2+z^2)+xyz(x+y+z)$$


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# Fubini's theorem problem

Let $f$ be a non-negative measurable function in $\mathbb{R}$. Suppose that $$\iint_{\mathbb{R}^2}{f(4x)f(x-3y)\,dxdy}=2\,.$$ Calculate $\int_{-\infty}^{\infty}{f(x)\,dx}$.

My first thought was to change the order of integration so that I integrate in $y$ first, since there's only a single $y$ in the integrand... but I'm not sure how/if that even helps me.

Then the more I thought about it, the less clear it was to me that Fubini's theorem even applies as it's written. Somehow I need a function of two variables. So should I set $g(x,y) = f(4x)f(x-3y)$ and do something with that? At least Fubini's theorem applies for $g(x,y)$, since we know it's integrable on $\mathbb{R}^2$. .... Maybe?

I'm just pretty lost on this, so any help you could offer would be great. Thanks!

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$g(x, y)$ is integrable on $\mathbb{R}^2$: $f$ is non-negative, so $|g|=g$. You know the integral of $g$ over $\mathbb{R}^2$ and it is finite. –  Vitaly Lorman Mar 10 '11 at 17:25
if both functions are integrable you can use Tonelli's theorem which is a simpler version of Fubini's theorem. I think for Tonelli to apply you don't have to know that $g(x,y)$ is integrable, you can just exchange the integration order. –  Rudy the Reindeer Mar 10 '11 at 17:45

Your thought is a good one. If you integrate in $y$ first you can change variable to $u=x-3y$. You are considering $x$ a constant, so you get a decoupled product of integrals of $f$ over the real line.

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Thanks so much. Chose your answer because you left more of it up to me =) –  Bey Mar 14 '11 at 20:39

I think both Vitali and Matt are right. As soon as $G(x,y)$ is integrable on $\mathbb{R}^2$, $$\iint_{\mathbb{R}^2}{G(x,y)\,dxdy}=\int_{\mathbb{R}}dx\,\int_{\mathbb{R}}{G(x,y)\,dy}=\int_{\mathbb{R}}dy\,\int_{\mathbb{R}}{G(x,y)\,dx}$$

So you can substitute: $$u=4x$$ $$v=x-3y$$ $$x=\frac{u}{4}$$ $$y=\frac{u}{12}-\frac{v}{3}$$ Jacobian $$J=\det D(x(u,v),y(u,v))=\begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}=\begin{vmatrix}\frac{1}{4} & 0\\\\\frac{1}{12} & -\frac{1}{3}\end{vmatrix}=-\frac{1}{12}$$

Then $$\iint_{\mathbb{R}^2}{f(4x)\,f(x-3y)\,dxdy}=-\frac{1}{12}\int_{-\infty}^{\infty}\,du\,\int_{\infty}^{-\infty}{f(u)\,f(v)\,dv}=\frac{1}{12}(\int_{-\infty}^{\infty}{f(z)\,dz})^2=2$$ Thus giving us $$\int_{-\infty}^{\infty}{f(z)\,dz}=2\sqrt 6$$

Please tell me if I $\mathbb{F}\bigoplus$ something up.

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Where did $a$ and $b$ come from? I think you are out a factor 4 as in the next to last line as $\int_{-\infty}^{\infty}f(4x)dx=\frac{1}{4}\int_{-\infty}^{\infty}f(x)dx$ and the same for 3 in the $y$ direction. Then before taking the square root, you multiply by 48(should be 12) instead of dividing, getting $\sqrt{24}i$. –  Ross Millikan Mar 10 '11 at 21:56
1. Sorry. (a,b)=(u,v). I fixed that already. –  Viktor Mar 10 '11 at 22:01
When you change variables, you should multiply the absolute value of the determinant. –  Byron Schmuland Mar 10 '11 at 22:02
It should be $y=\frac{u}{12}-\frac{v}{3}$ This will fix the factor $4$. –  Ross Millikan Mar 10 '11 at 22:04
Thanks Byron! Thanks Ross! Haven't done that for awhile. –  Viktor Mar 10 '11 at 22:09


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# latex???

hi,
i want to typeset my page into  landscape  mode  in latex.... instead of the default portrait mode... how do i do it .....??
i tried \landscape it would not work ......
pl help i need it urgently ......
subhro...
###### Who is Participating?

Commented:
That should work, but it may depend on the type of document you are doing (article, slide ...)

Another way could be to set the paper size and margins directly, like this:

\setlength{\textheight}{8.75in}
\setlength{\textwidth}{6.8in}

Here you put the size you need (letter landscape would be 11x8.5)

And you can change the margins if you wish too

\setlength{\topmargin}{0.25in}
\setlength{\oddsidemargin}{-.19in}
\setlength{\evensidemargin}{-.19in}
0

Author Commented:
with some changes i was able to do it ... thanx
0
Question has a verified solution.

Are you are experiencing a similar issue? Get a personalized answer when you ask a related question.

Have a better answer? Share it in a comment.


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# Wurtz Reaction with acidic alkyl halides [closed]

My chemistry book says we cannot use alkyl halides containing acidic hydrogen for Wurtz reaction. Why is this so?

## closed as off-topic by Mithoron, a-cyclohexane-molecule, aventurin, Todd Minehardt, TyberiusAug 30 '18 at 14:49

This question appears to be off-topic. The users who voted to close gave this specific reason:

If this question can be reworded to fit the rules in the help center, please edit the question.

• The intermediate is an alkyl sodium, a highly basic species which will be quenched by any proton source. Acidic hydrogens will also react directly with the sodium metal. – Waylander Aug 22 '18 at 11:57

The Wurtz reaction involves the reaction of alkyl halide with sodium in dry ether. In this reaction, two molecules of alkyl halide react to form a alkyl-alkyl chain and sodium halide is also produced -

$$\ce{R-X + 2Na + X-R ->R-R + 2NaX} \tag{\ce{X} = halogen}$$

Now, sodium is a strongly electropositive metal, and is above hydrogen in the electrochemical series. Thus it replaces acidic hydrogen wherever it occurs. For example,

$$\ce{CH3COOH + Na ->CH3COO- Na+ + 1/2 H2}$$

So, sodium reacts with the acidic hydrogen present in the starting alkyl halide.

Also, acid-base reactions are, in general, faster than a substitution reaction. If there was no acidic hydrogen, alkyl sodium intermediate would have formed and an SN2 reaction would have taken place.

However, in this case, if any alkyl sodium intermediate is formed, it would accept the acidic hydrogen and convert into an alkane, stopping the reaction.

$$\ce{R- Na+ + H-M -> R-H + NaM}$$

(R = alkyl group, M represents the functional group or chain attached to the acidic hydrogen)


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# Can I use TikZ (instead of PSTricks) for latex2html5?

I found this today:

http://latex2html5.com/

Author interactive math equations and diagrams online using LATEX and PSTricks

It looks nice, but I'm more familiar with TikZ. Is anybody aware of something equivalent which is not based on PSTricks but TikZ?

-
you can use tex4ht for this job - tex.stackexchange.com/q/35145/2891 for tex4ht + tikz, albany.edu/~hammond/demos/Html5/arXiv for tex4ht + mathjax + html5 –  michal.h21 Nov 18 '13 at 10:03
You need to know that not all PSTricks constructs are currently supported. The author probably chose PSTricks because it is much less verbose compared to TikZ. Parsing TikZ is much more complicated. –  Weirdstress Function Nov 18 '13 at 14:29
@michal.h21 Do you mind cooking up the simplest example with tex4ht? –  percusse 6 hours ago


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chapter  26
4 Pages

## Three characteristic drop size scales [46]

A drop is an object of an intermediate size: from molecular associates to macrodrops of a new phase. The physical basis for changing concepts, in fact, is absent. The nature of the intermolecular potential acting in the system of molecular associates implies the absence of saturation and, as a consequence, the absence of a limit for the degree of association [40]. When describing the equilibrium drop there is no Laplace equation, which greatly simplifies mathematical analysis. Let us consider the behaviour of the surface tension on the

drop size σ(R). Three characteristic sizes can be distinguished which correspond to: 1) the value of the radius Rb, the greater of which the surface tension values are close to the volume value, with small curvatures of the surface of large drops; 2) the value of the radius Rt above which the thermodynamic description of the surface tension of the drop is justified, when both the discrete nature of the matter and the contributions of spontaneous fluctuations can be neglected; and 3) the critical value R0 of the beginning of the formation of the phase associated with the appearance of the surface tension of the dense phase.


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Journal cover Journal topic
Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
Journal topic
Hydrol. Earth Syst. Sci., 23, 3933–3944, 2019
https://doi.org/10.5194/hess-23-3933-2019
Hydrol. Earth Syst. Sci., 23, 3933–3944, 2019
https://doi.org/10.5194/hess-23-3933-2019

Research article 25 Sep 2019

Research article | 25 Sep 2019

# Global-scale human pressure evolution imprints on sustainability of river systems

Global-scale human pressure evolution imprints on sustainability of river systems
Serena Ceola1,*, Francesco Laio2, and Alberto Montanari1 Serena Ceola et al.
• 1Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, Bologna, 40136, Italy
• 2Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Turin, 10129, Italy
• * Invited contribution by Serena Ceola, recipient of the EGU HS Division Outstanding Early Career Scientist Award 2019.

Correspondence: Serena Ceola (serena.ceola@unibo.it)

Abstract

Human pressures on river systems pose a major threat to the sustainable development of human societies in the twenty-first century. Previous studies showed that a large part of global river systems was already exposed to relevant anthropogenic pressures at the beginning of this century. A relevant question that has never been explained in the literature so far is whether these pressures are increasing in time, therefore representing a potential future challenge to the sustainability of river systems. This paper proposes an index we call “Differential Human Pressure on Rivers” (DHPR) to quantify the annual evolution of human pressure on river systems. DHPR identifies a per-year percentage increment (or decrement) of normalized human pressures on river systems (i.e., ratio of annual values to long-term average). This index, based on annual nightlights and stationary discharge data, is estimated for 2195 major river basins over a period of 22 years, from 1992 to 2013. The results show that normalized annual human pressure on river systems increased globally, as indicated by an average DHPR value of 1.9 % per year, whereby the greatest increase occurred in the northern tropical and equatorial areas. The evaluation of DHPR over this 22-year period allows the identification of hot-spot areas, therefore offering guidance on where the development and implementation of mitigation strategies and plans are most needed (i.e., where human pressure is strongly increasing).

1 Introduction

The detrimental effects of anthropogenic activities on river systems have been reported extensively . Increasing anthropogenic activities, demographic expansion and the improvement of living standards are threatening sustainability at a global level. The development of human activities and associated human pressure on river systems often results in the emergence of threats to water security, from both human and river biodiversity perspectives . There is an urgent need to manage and guide this development by strategies that consider how human pressure on river systems can be sustainable in the long term.

Human–water interactions controlling human pressure on river systems should be analyzed at a detailed level, based on local knowledge of the spatial distribution of river discharge, human population and associated interrelationships . In order to compare, identify and prioritize areas of high human pressure across the globe, estimates of human pressures should be assessed at global scale. Understanding the spatial and temporal patterns in human pressure on river systems is fundamental for the development and implementation of targeted strategies.

A first high-resolution global-scale assessment of human pressure on river systems showed severe water threat levels for nearly 4.8 billion people. In that study, the most threatened areas were located across the United States, Europe, central Asia, India, eastern China and the Middle East. While that analysis showed the relevance and extent of the problem and the need to take action, it is now important to understand how this issue developed and how it is likely to progress in time. Only when the historic development and the key factors that triggered it are understood can reality-based targets be established and sound strategies be developed. In this context, recent efforts focusing on the terrestrial system provided updated estimates of the global human footprint monitoring changes from 1993 to 2009 .

Powerful tools are now available to carry out this analysis. Earth system modeling and remote sensing observations have produced global datasets that provide unprecedented possibilities to analyze and identify human pressures on river systems, as well as their progress in time. There is information on human presence and activity (EC-JRC2015; CIESIN2018; NOAA2017), river and watershed delineation , river discharge , and water-related threats that can be analyzed independently or combined to provide insight on human–water interactions.

We propose a methodology for analyzing and mapping the historic evolution of human pressure on rivers. Given a river site and its contributing area, human pressure on river systems is defined as the ratio between the cumulative human presence and activity across the contributing area and the natural discharge generated within the same contributing area. We estimate human presence and activity, which is mainly linked to population density and level of development, by analyzing nightlights, retrieved from satellite images monitoring nocturnal luminosities. Natural discharge values, which epitomize surface hydrological processes within a river basin, are computed from runoff data. Nightlights and river runoff are selected as the key variables to calculate human pressure on river systems because of the availability of valuable global-scale and spatially explicit data that allows the analysis over time. More specifically, we compute human pressure on river systems on an annual basis, where human presence and activity varies across years, while natural discharge is assumed to be stationary during the study period. We therefore take the catchment as the spatial entity whereon human pressure on river systems is evaluated and then measure its annual evolution by defining the Differential Human Pressure on Rivers (DHPR) index. DHPR is computed as the annual time derivative of human pressure on river systems and identifies a per-year percentage increment (or decrement) of normalized human pressures (i.e., ratio of annual values to long-term average). Global-scale DHPR values are estimated for the period 1992–2013 for 2195 major river basins. DHPR can be used to identify hot-spot areas of change where there is a marked and consistent increase of human pressure on river systems over time. This is valuable information on which a robust planning strategy can be based, targeting actions to address water threats in key areas and bearing important implications for a sustainable development of human societies close to river systems in the near future. To prove the validity of our methodology, the relationship between human pressure on river systems and existing datasets, i.e., water threats and the terrestrial human footprint , is investigated. Global values of human pressure are contrasted with the corresponding water threat and human footprint values.

The paper is organized as follows. In Sect. 2 we describe the data and the methodology developed for the estimation of global-scale DHPR values, including also the correlation analysis with alternative datasets. The main outcomes are reported in Sect. 3. Results are then discussed in Sect. 4, including also some concluding remarks.

2 Materials and methods

## 2.1 Global-scale river network and runoff data

The Simulated Topological Network STN-30 was the digital river network used in this work. The STN-30 river network originates from a 0.5 flow direction grid (i.e., nearly 55 km at the Equator) and offers many different river attributes, such as drainage area, river length, distance to river outlet and river basin delineation. We compute average annual natural river discharge as derived from the Global Composite Runoff Fields dataset , which provides long-term mean annual runoff data along the STN-30 river network. Natural river discharge was derived from a routing scheme based on flow direction paths along the STN-30 as follows (see Figs. 1b and S1c in the Supplement):

$\begin{array}{}\text{(1)}& {Q}_{i}=\sum _{j=\mathrm{1}}^{{N}_{i}}{R}_{j}\cdot {A}_{j},\end{array}$

where Qi (km3 yr−1) is the long-term mean annual discharge in any grid cell i; Rj (km yr−1) and Aj (km2) are the long-term mean annual specific runoff (i.e., per unit area) and the area of grid cells j, respectively. The identifier Ni is coincident with the number of upstream grid cells j.

Figure 1Estimation of human pressure on river systems by explicitly considering the topological nature of river basins. Schematic illustration of (a) cumulative human presence and activity, HP (–), calculated as routed nightlight values for each grid cell at 0.5 resolution, obtained from downstream propagation following flow direction paths (grey line; see Eq. 3), (b) natural river discharge, Q (km3 yr−1), obtained from downstream propagation following flow direction paths (grey line; see Eq. 1), (c) standardized human pressure on river systems, F(t) (–), for each grid cell (see Eqs. 2 and 4, where min(fk)=50 km−3 yr and max(fk)=30 000 km−3 yr).

## 2.2 Global-scale data on human presence and activity

There are several possibilities for the estimation of human presence and activity at high spatial resolution globally. Traditional datasets that provide gridded data on population densities and/or gross domestic product (GDP) estimates could be employed. The Gridded Population of the World, developed by the Center for International Earth Science Information Network at Columbia University, and the Global Human Settlement Layer, provided by the European Commission, Joint Research Centre, are the most acknowledged datasets on population densities. The Gridded Population of the World dataset provides population counts and densities for the years 2000, 2005, 2010, 2015 and 2020 (extrapolated from 2010 population data) at 0.0083 resolution (i.e., nearly 1 km at the Equator) for the entire globe. However, constant population densities are provided within each census unit, thus resulting in a significant limitation for the proposed analysis. Differently, the Global Human Settlement Layer dataset provides a spatial variability within censuses at either 1 km or 250 m spatial resolution. However, this dataset offers a temporal evolution of population densities for the years 1975, 1990, 2000 and 2015 estimated from an exponential growth model that uses limited ground-based data. Concerning GDP estimates, a temporal sequence of gridded datasets at the global scale is not available to date. Yearly country-based GDP values are usually provided, even though some developing countries may present low-quality statistical data.

An alternative dataset is represented by nightlights, which overcomes some major limitations of the aforementioned ones. Nightlights are used in this work as a proxy for human presence and activity, being the best available option to analyze and map the evolution of global-scale human pressure on river systems both in space and time. Nightlights are satellite data derived from the United States Air Force Defense Meteorological Satellite Program (DMSP). The satellites are equipped with Operational Linescan System (OLS) sensors, and the information collected is freely provided by the National Oceanic and Atmospheric Administration's National Geophysical Data Center (NOAA2017). Stable light composites, available as raster products, are employed here. They monitor the global distribution of nocturnal luminosities associated with anthropogenic activities. More specifically, nightlights provide combined information on human presence and economic activities, as high luminosity can refer to either highly populated or major capital investment areas. Nightlights have been extensively employed as a proxy for human presence and activity for several purposes such as population , urban and poverty mapping , flood risk , economic analysis , and light pollution .

Nightlight values are expressed as digital numbers (DNs) that range from 0 (pitch dark areas) to 63 (brightest areas), and they are produced on a yearly basis from 1992 to 2013 (i.e., each pixel shows an average luminosity within a year). Nightlights are provided at 0.00833 spatial resolution and cover areas within 75 N and 65 S, 180 W and 180 E. Six different satellites collected nightlight data during the observation period, with overlapping satellites during some years (from 1997 to 2007). In the case of multiple satellites operating simultaneously, an average value between the two simultaneous satellites was computed in order to obtain unique yearly nightlight values. Since nightlight products are not calibrated on board, a well-established intercalibration procedure was performed , before computing human pressure on river systems.

## 2.3 Historic evolution of global-scale human pressure on river systems: computational steps

The Differential Human Pressure on Rivers (DHPR) index was derived from the analysis of the historic evolution of annual values of human pressure on river systems. The computational steps explicitly incorporate catchment topology and use a routing scheme based on flow directions to evaluate the downstream accumulation of human presence and activity and natural river runoff.

Given a river site and its contributing upstream area, annual values of human pressure on river systems fi(t) (km−3 yr) were defined as the ratio between the cumulative human presence and activity across the contributing area HPi(t) (–) and the long-term average natural river discharge generated within the same area Qi (km3 yr−1); see Eq. (1). Namely,

$\begin{array}{}\text{(2)}& {f}_{i}\left(t\right)=\frac{{\mathrm{HP}}_{i}\left(t\right)}{{Q}_{i}},\end{array}$

where i identifies a generic river network grid cell and t represents the study year, from 1992 to 2013. The term “cell” below always refers to 0.5 by 0.5 grid cell, as defined by the STN-30 river network employed here . Overall, data from 20 770 grid cells for 2195 river basins were used. The cumulative human presence and activity HPi(t) (–) in any grid cell i was calculated from contributing upstream cells as routed nightlight values (see Figs. 1a and S1a and b):

$\begin{array}{}\text{(3)}& {\mathrm{HP}}_{i}\left(t\right)=\sum _{j=\mathrm{1}}^{{N}_{i}}{\mathrm{NL}}_{j}\left(t\right),\end{array}$

where NLj(t) represents summed nightlight values in grid cells j for year t (i.e., the value of pixels for nightlights at 0.00833 resolution were summed to 0.5 by 0.5 grid cells) and Ni is the number of upstream cells j. Grid cells with null nightlight data throughout the whole study period were discarded.

In order to estimate the historic evolution of human pressure on river systems across the entire globe, annual values of fi(t) were first standardized to a dimensionless 0–1 scale Fi(t) as follows (see Fig. 1c):

$\begin{array}{}\text{(4)}& {F}_{i}\left(t\right)=\frac{{f}_{i}\left(t\right)-\mathrm{min}\left({f}_{k}\right)}{\mathrm{max}\left({f}_{k}\right)-\mathrm{min}\left({f}_{k}\right)},\end{array}$

where k identifies a generic grid cell where fk is either the absolute minimum or maximum value of fi(t) across all considered grid cells and years. The long-term average of standardized human pressure on river systems $\stackrel{\mathrm{‾}}{{F}_{i}}$ was then computed as the mean of annual values from 1992 to 2013. Standardization was essential to test the reliability of the proposed methodology. Indeed, standardized human pressure values were contrasted with existing well-acknowledged datasets, i.e., water threats and the terrestrial human footprint , by performing a regression analysis (see Sect. 2.4). The historic evolution of standardized human pressure values was assessed by performing the following linear regression:

$\begin{array}{}\text{(5)}& \frac{{F}_{i}\left(t\right)}{\stackrel{\mathrm{‾}}{{F}_{i}}}={a}_{\mathrm{0}}+{\mathrm{DHPR}}_{i}\cdot t+\mathit{ϵ}\left(t\right),\end{array}$

where DHPRi (yr−1) is the Differential Human Pressure on Rivers index in grid cell i, a0 is the intercept, and ϵ(t) represents regression residuals. DHPR represents the relative change rate (i.e., a percentage increment or decrement) of normalized human pressure values, defined as the ratio between annual and long-term average standardized values of human pressure. We employed normalized human pressure values to easily compare relative change rates at the global scale. Positive (or negative) DHPR values correspond to increasing (or decreasing) trends of human pressure on river systems in the study period. Null DHPR values identify a time-invariant behavior (i.e., significant changes in time are not detected). For example, a value of DHPR equal to 4 % per year represents the condition for which normalized human pressure on river systems increases on average every year of 4 % with respect to its initial condition (i.e., in 1992). Thus in the 22-year study period, there is an 88 % relative increment of ${F}_{i}\left(t\right)/\stackrel{\mathrm{‾}}{{F}_{i}}$ values. Without any normalization, change rates of human pressure would have been proportional to standardized values Fi(t), which clearly depend on the contributing area, nightlight values and river discharge data. By computing a relative increment (or decrement) of human pressure on river systems, the DHPR index is expected to be a valuable tool, particularly for scarcely illuminated regions. P values from Student's t test and coefficients of determination R2 were computed for each river basin to test the statistical significance of the linear regression given by Eq. (5).

## 2.4 Correlation analysis between estimates of human pressure on river systems and existing global-scale datasets

In order to test the reliability of the proposed approach, we compared our estimates of human pressure on river systems with well-acknowledged datasets based on different methodologies. Two alternative datasets were used for this comparison: (i) water threats to human water security and river biodiversity and (ii) human footprint .

Water threat values, defined by , identify either human water security or river biodiversity threats and represent the level of endangerment of river systems. Estimates of water threats derive from a routing scheme based on a combination of 23 drivers normalized by the natural river discharge. Global-scale water threat data are provided as a map on the status of river systems at a given location and at a particular point in time, around the year 2000, with a 0.5 resolution. For our correlation analysis we used gridded water threat data, freely available at http://www.riverthreat.net/index.html (last access: 8 March 2017). Although human water security and river biodiversity threats are strongly correlated, in this paper we opted to examine both threats.

Human footprint estimates, developed by , provide a measure of human pressure on the environment. Human footprint considers the entire terrestrial realm and not only river systems. Human footprint values on land areas are derived from eight different proxy variables for human pressure. Normalized cumulative values identify the percentage of relative human influence within a region. Global-scale gridded data of human footprint are available for years 1993 and 2009 at 0.00833 resolution (https://doi.org/10.5061/dryad.052q5; see also ). For our correlation analysis, we used data from both years and averaged the value of pixels for human footprint at 0.00833 to 0.5 by 0.5 grid cells.

Standardized values of human pressure on river systems Fi(t) computed in any grid cell i were contrasted with water threat and human footprint data. This relationship was investigated by performing the following log-linear regression analysis:

$\begin{array}{}\text{(6)}& {\mathrm{log}}_{\mathrm{10}}{F}_{i}\left(t\right)={m}_{X}\cdot {X}_{i}\left(t\right)+{b}_{X}+{\mathit{ϵ}}_{X},\end{array}$

where Xi(t) identifies either water threat or human footprint values in grid cell i, mX represents the slope of the log-linear regression line, bX is the intercept and ϵX represents regression residuals. Water threat values were contrasted with standardized human pressure estimates for year 2000, while human footprint data were compared against 1993 and 2009 estimates. We computed P values from Student's t test and the coefficient of determination R2 to check the statistical significance of the regression analysis. A statistically significant correlation in space between our estimates of human pressure on rivers and previous metrics warrants the use of the proposed variable as a valuable alternative. In addition, given the availability of time series, it allows the quantification of changes in time.

## 2.5 Regionalization of the Differential Human Pressure on Rivers index

The Differential Human Pressure on Rivers (DHPR) index describes the historic evolution of human pressure on river systems driven by the heterogeneity of anthropogenic activities and hydrological and climatic regimes. DHPR values computed for the considered 2195 river basins were analyzed by considering aggregated spatial regions delineated by river basins with similar annual average runoff and temperature. The regions considered were hydrobelts and hydroregions , which incorporate key hydraulic and climatic features driving natural river discharges. Hydrobelts (8 in total, i.e., boreal, northern midlatitude, northern dry, northern subtropical, equatorial, southern subtropical, southern dry, southern midlatitude) are classified by maximizing the differences among belts and minimizing the variability within belts; hydroregions (26 in total) are hydrobelts decomposed on a continental basis.

3 Results

Annual values of standardized human pressure on river systems F(t) and the long-term average $\stackrel{\mathrm{‾}}{F}$ were calculated for 20 770 grid cells distributed across 2195 river basins over the period 1992–2013 and then grouped by region. The regions considered (i.e., hydrobelts and hydroregions; Meybeck et al.2013) are shown in Fig. 2. Globally, long-term average standardized human pressures $\stackrel{\mathrm{‾}}{F}$ presented a considerable heterogeneous spatial pattern (Fig. 2), which is to be expected due to the intrinsic variations in the considered drivers (Fig. S1). Standardized human pressure values, ranging from 0 to 1, depend on the spatial extent of the contributing area, the level of human presence and activity, as derived from nightlights, and natural discharge values. As a result, large human pressures are typical of river basins with low natural discharges and high human presence and activity. Conversely, low human pressures are generally found across river basins with little human presence and activity and high river discharge. Accordingly, high $\stackrel{\mathrm{‾}}{F}$ values were found in the northern midlatitudes and subtropical regions (i.e., eastern United States, Europe, India and eastern China), whereas low scores were typical of boreal (i.e., northern Russia) and equatorial (i.e., central Brazil and central Africa) areas. Estimates of standardized human pressure could be analyzed by ranking river basins as a decreasing function of either discharge or human presence and activity. When examining the first 15 basins, ranked from largest to smallest natural river discharge (see Table A1), we found $\stackrel{\mathrm{‾}}{F}$ values lower than the 90th percentile (${\stackrel{\mathrm{‾}}{F}}_{\mathrm{90}}=\mathrm{0.426}$). In particular, 11 out of 15 river basins showed $\stackrel{\mathrm{‾}}{F}<{\stackrel{\mathrm{‾}}{F}}_{\mathrm{50}}\left(=\mathrm{0.055}\right)$, and 13 out of 15 with $\stackrel{\mathrm{‾}}{F}<{\stackrel{\mathrm{‾}}{F}}_{\mathrm{75}}\left(=\mathrm{0.201}\right)$. This result was expected, since for larger discharges and assuming equal levels of human presence and activity, lower estimates of standardized human pressure can be found. Aggregation proved that the boreal, northern midlatitude and equatorial hydrobelts were equally represented, with 4 river basins each in the first 15. High natural discharges are indeed typical of these three hydrobelts. When ranking river basins as a decreasing function of human presence and activity (see Table A2), we found that 11 out of the first 15 basins were located within the northern midlatitude belt, which is known to be the most populated hydrobelt across the globe. Estimates of human presence and activity within a river basin provide embedded and combined information about the extension of the contributing area, the total population and the economic activity in that area. This estimate is not directly proportional to the river basin area. Indeed, Amazon, the river basin with the largest drainage area in the world, is not among the 15 largest river basins based on human presence and activity. Large estimates of human presence and activity do not necessarily correspond to high levels of standardized human pressure, but overall 13 out of 15 river basins showed $\stackrel{\mathrm{‾}}{F}>{\stackrel{\mathrm{‾}}{F}}_{\mathrm{50}}$.

Figure 2Global distribution of long-term average standardized human pressure on river systems, $\stackrel{\mathrm{‾}}{F}$. Colors differentiate grid cells (20 770 in total) according to levels of $\stackrel{\mathrm{‾}}{F}$. Hydrobelt boundaries are also shown (i.e., boreal in purple, northern midlatitude in dark blue, northern dry in cobalt, northern subtropical in light blue, equatorial in green, southern subtropical in grey, southern dry in brown, southern midlatitude belt in black).

To test the reliability and consistency of the proposed methodology, standardized values of human pressure on river systems F(t) were contrasted with well-acknowledged and consolidated datasets mapping human pressure on terrestrial and freshwater systems. Overall, a consistent worldwide behavior emerged, supported by statistically significant relationships among indices (Table S1 in the Supplement). High scores for standardized human pressure on river systems correlate well with high values of both water threat and human footprint, clearly implying severe endangerment levels. A better correlation was found with water threats, rather than human footprint values. This was expected, since human footprint considers the entire terrestrial realm and does not exclusively focus on river systems. Global and regional results presented a fair data scatter, which reduced when focusing at smaller spatial scales. Our approach based on nightlight and river discharge data cannot explain and totally embed the geographical heterogeneity and the variability of human–water interactions. However, it represents a first step forward in mapping the historic development of human pressure and, by focusing on river systems, complements recent outcomes on the terrestrial environment (i.e., human footprint; Venter et al.2016a, b).

In order to identify priorities and hot-spot regions of change, and thus produce consistent and reliable blueprints to manage human pressure on river systems, it is fundamental to analyze its historic evolution and identify areas where human pressure is increasing. The Differential Human Pressure on Rivers (DHPR) index was calculated at the outlet of the considered 2195 river basins over the period 1992–2013. The global analysis of DHPR revealed positive change rates (Fig. 3), with values within −0.4 %–3.7 % per year (lower and upper quartiles, mean = 1.9 % per year; see Fig. 4). Results at the basin scale showed an heterogeneous spatial distribution of change rates, confirming and complementing recent country-based outcomes . Overall, markedly positive DHPR values were found across river basins with low to moderate standardized human pressure on river systems, whereas regions with high standardized human pressure showed either slightly negative or negligible changes in time. Table 1 reports DHPR estimates for 15 major river basins across the globe. Values for all the considered river basins are provided in Table S2.

Figure 3Global distribution of the Differential Human Pressure on Rivers index, DHPR. The map shows the relative change rate (i.e., per-year percentage increment or decrement) of normalized human pressure values, defined as the ratio between annual and long-term average standardized human pressure on river systems, $F\left(t\right)/\stackrel{\mathrm{‾}}{F}$. Colors differentiate basins (2195 in total) according to levels of DHPR. Insets show the historic evolution of normalized human pressure on river systems for Amazon, Danube, Nile and Chang Jiang river basins, where data are depicted as black dots and the red lines represent Eq. (5).

Table 1Estimates of the Differential Human Pressure on Rivers index, DHPR, for 15 major river basins. Basin name, contributing area (106 km2), hydrobelt, hydroregion, DHPR values (% per year), coefficient of determination R2, and P values for Student's t test are reported.

Individual catchment-scale results were aggregated at the regional level. Grouping results by hydrobelt and hydroregion was a meaningful way to perform this spatial aggregation. In fact, hydrobelts and hydroregions incorporate key hydrologic and climatic features driving average discharge regimes (Figs. 4 and S2). The boreal belt identifies areas with average annual temperatures below 0 C . Within this belt, we found river basins characterized by high natural discharges and a limited human presence and activity, resulting in low values of standardized human pressure. River basins in the boreal belt showed negative DHPR scores, with lowest values across Canada, Europe and eastern Siberia. Northern latitude belts (midlatitude, dry and subtropical) are known to be the most populated areas across the globe. Basins within northern latitude belts presented a wide range of natural discharges, showing fairly high values of standardized human pressure. When looking at the historic evolution of standardized human pressure on river systems, the northern midlatitude belt (i.e., corresponding to the most developed countries in the Northern Hemisphere) showed DHPR values almost symmetrically distributed around zero, with positive change rates across Europe and Asia and negative trends across the United States. The highest DHPR estimates were found in river basins located within the northern dry, northern subtropical and equatorial belts, in particular across Africa and Asia. River basins located across southern latitude belts (subtropical, dry and midlatitudes) typically presented positive DHPR values, with slightly smaller change rates compared to northern dry, subtropical and equatorial belts.

Figure 4Regional patterns of the Differential Human Pressure on Rivers index, DHPR. Box plot statistics for DHPR grouped by hydrobelt : world (2195 basins), boreal (170 basins), northern midlatitude (800 basins), northern dry (133 basins), northern subtropical (319 basins), equatorial (354 basins), southern subtropical (141 basins), southern dry (84 basins), southern midlatitude (194 basins). Box plots include the median (thick black line), interquartile range (colored boxes) and whiskers (confidence interval of ±2.7σ).

DHPR identifies critical zones where increasing trends in human pressure on river systems may undermine human security and sustainable development in the near future. Hot-spot areas of change are represented by river basins located within the northern subtropical and equatorial belts across Africa and Asia, showing markedly positive change rates in the 1992 to 2013 period (Figs. 3, 4 and S2). Future climate change scenarios and demographic projections will impact on future DHPR values. For instance, when considering African basins across the northern subtropical and equatorial belts, markedly positive DHPR scores are likely to be expected, with population and socioeconomic level predictions playing a major role compared to changes in natural river discharge . Indeed, the highest population growth rates are predicted to be in Africa, where more than half of the global population increase (i.e., nearly 83 million people by year) will settle by 2050 .

4 Discussion and conclusions

Human development and riverine ecosystems intimately depend on the geographic and temporal distribution of natural river discharge , whose global-scale pattern is primarily controlled by hydrogeomorphologic and climatic drivers. Current human pressures on river systems are likely to be affected by future population increases and climate change . As a consequence, managing the development of human societies and protecting fluvial ecosystems will be a significant challenge . The Differential Human Pressure on Rivers (DHPR) index proposed here is a simple tool that analyzes for the first time the temporal evolution of human pressure on river systems. DHPR identifies hot-spot areas of change, offering guidance on where to implement mitigation strategies. On account of its simplicity and the availability of spatially and temporally explicit data for its calculation, DHPR is a valuable alternative and a step forward to well-acknowledged and consolidated datasets mapping human pressure on terrestrial and freshwater systems (see Table S1).

We acknowledge that the proposed index presents some limitations. Our approach defines human pressure on rivers as a basin-scale cumulative effect of residing population and its economic activities on the natural river discharge at the basin outlet. In other words, we focus on (1) how many people live and act on a river basin (namely, the sum of nightlights) and (2) in which way this anthropogenic effect is diluted with river discharge. Local aspects such as dams and water withdrawals for civil, industrial and irrigation purposes are not taken into account. Therefore, nightlights and river discharge are considered the sole controlling and the best representative drivers of human pressure on rivers. Nightlights have been proven to be an effective tool monitoring human presence and activity, although they feature several potential weaknesses . The low resolution of nightlight sensors may cause zero values of nightlights in populated areas. The limited radiometric range may result in saturated nightlight values in urban areas (i.e., saturation effect) or in larger lit areas (i.e., blooming effect). To dampen these effects, estimates of human pressure on river systems were first standardized and then normalized. In addition, light pollution abatement strategies , employed to reduce the artificial sky brightness and preserve world's ecosystems , can cast some doubt on nightlight values and on their evolution in time. However, these uncertainties, which should be treated with caution when analyzing small areas, are barely detectable at the basin scale employed here.

Natural river discharge, as computed from Eq. (1), is defined as a function of hydrological and geomorphological variables within a river basin. One major limitation of the proposed methodology is that the variability of natural river discharges within and between years is not considered. Since the scope of the present study is to analyze and map the annual evolution of human pressure on river systems at global level, intra-annual variability is not of interest. Concerning interannual variability, future studies, focused on local-scale problems and smaller areas, are planned to embed discharge variability between years and thus account for hydrological changes. Furthermore, the proposed approach relies on an ideal case where groundwater fluxes and anthropogenic factors (i.e., water intakes, transboundary water management and environmental flow requirements), which may potentially affect human pressure on river systems, are not taken into account. If considered, environmental flow requirements would reduce the natural river discharge (i.e., on an annual basis, 80 % of natural discharge is allocated as environmental flow; Mekonnen and Hoekstra2016). Consequently, human pressure on river systems based on environmental flow requirements would result in higher absolute values, but change rates, expressed by DHPR, would be equal to change rates of human pressure based on natural discharges. Similarly, one could account for groundwater fluxes. If considered, groundwater would potentially enhance water availability, thus resulting in lower absolute values of human pressure. However, this issue, which involves a sustainable use of groundwater resources, is beyond the scope of the proposed analysis.

Our approach, estimating the historic evolution of human pressure on river systems, explicitly considers the connectivity and the structure of the river network and provides simplified information about interactions among hydrological, geomorphological and human variables within a river basin. DHPR estimates allow a spatially and temporally explicit analysis of human–water interactions at the global scale. Our study identifies critical zones where the change rate of human pressure may undermine human security and sustainable development in the near future. The simplicity of the proposed methodology for assessing human pressure on river systems and the ease with which it can be used to reconstruct historic series makes it a powerful tool to be used independently or to be incorporated into a planning framework, targeting actions to address water threats in key areas.

Code and data availability
Code and data availability.

Nightlight data used for this article are available at NOAA Earth Observation Group (http://ngdc.noaa.gov/eog/dmsp/downloadV4composites.html, ). Water threat data were downloaded from http://www.riverthreat.net/index.html (). Human footprint data are available at https://doi.org/10.5061/dryad.052q5 (). Data on human pressure on river systems are derived from elaborations through a non-open-source code developed by the authors. More information can be gathered upon e-mail request to serena.ceola@unibo.it.

Appendix A

Table A1Estimates of long-term average standardized human pressure on river systems $\stackrel{\mathrm{‾}}{F}$. River basins are ranked as a decreasing function of natural river discharge, Q. The first 15 river basins are examined. Basin name, hydrobelt, hydroregion, $\stackrel{\mathrm{‾}}{F}$ and quantile distribution, DHPR values (% per year), coefficient of determination R2, and P values for Student's t test are reported.

Table A2Estimates of long-term average standardized human pressure on river systems $\stackrel{\mathrm{‾}}{F}$. River basins are ranked as a decreasing function of human presence and activity, HP. The first 15 river basins are examined. Basin name, hydrobelt, hydroregion, $\stackrel{\mathrm{‾}}{F}$ and quantile distribution, DHPR values (% per year), coefficient of determination R2, and P values for Student's t test are reported.

Supplement
Supplement.

Author contributions
Author contributions.

SC, FL and AM designed the research. SC carried out the research, developed the model code and performed the simulations. SC prepared the manuscript with contributions from all co-authors.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

The authors gratefully thank Irene Soligno for modeling tool support.

Financial support
Financial support.

This research has been supported by the European Research Council (project CWASI, grant no. 647473).

Review statement
Review statement.

This paper was edited by Hubert H. G. Savenije and reviewed by Wouter Berghuijs and one anonymous referee.

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text:
## anonymous one year ago Explain how energy is transfered in a food web.

1. anonymous

@Aureyliant

2. anonymous

@Aureyliant

3. anonymous

4. Koikkara

Energy is transferred through the separate trophic levels of a food chain or web by feeding.

5. anonymous

yes but how is it transferred through a for web

6. Koikkara

well what do you know about food webs?

7. anonymous

not much

8. Koikkara

well, for example, lets take : grass - deer - lion when the deer eats the grass, around 10% of the energy in the grass will be transferred to the deer. Same thing happens when the lion eats the deer..i.e. 10% energy is transferred from deer to lion. is it right?

9. anonymous

yes

10. Koikkara

then, the longer the food chain, the lesser energy is remained at the end of the chain Also, Food web is made up of at least 2 food chains.

11. Koikkara

Now you know how energy is transferred through a web.

12. anonymous

but can you tell me the complete answer in like 2-3 sentences please :) @Koikkara

13. anonymous

@Koikkara

14. anonymous

@Aureyliant

15. anonymous

Explain how energy is transfered in a food web.

16. anonymous

We do not give out the direct answer. You need to figure it out yourself. @ Koikkara explained it perfectly to you. Summarize what he has said.

17. anonymous

ok but i need help summarizing it. I'm bad at that

18. anonymous

lets start with the first sentence.. it should go something like this right? Energy is transferred through the separate trophic levels of a food chain or web by feeding.

19. Koikkara

alrite, i will help then ?

20. Koikkara

When a animal in the food chain eats another lower in the food chain, the energy is transferred by heat and movement. Is it possible ?

21. Koikkara

srry...when "an animal"

22. Koikkara

@geny55 write both points together... you got a complete answer.

23. anonymous

ok so like this.. Energy is transferred through the separate trophic levels of a food chain or web by feeding. When a animal in the food chain eats another lower in the food chain, the energy is transferred by heat and movement. @Koikkara

24. Koikkara

$$\Large\rm\color{blue}{There~you~did~it~!!~Congrats~!}$$

25. anonymous

are you sure that the answer says how energy is transferred in a FOOD WEB? @Koikkara

26. Koikkara

ofcourse, why not... ? Means you didn't understand the topic...!! Doubts are welcome.

27. anonymous

no I'm just making sure.. i don't want to get it wrong

28. anonymous

@Koikkara

29. anonymous


text:
# RapidMiner

## Logistic Regression

Regular Contributor

# Logistic Regression

Hi,

I have 2 independent variables namely Score1 and Score2 and my dependent class variable is state which contain two values yes and No.My objective is to get
log(odd ratio) or logit =a+b*score1+c*score2. I applied logistic regression on these data and the result is like.

Bias(offset)=1.2
W[score1]=17.32
W[Score2]=18.33

Please anybody help me to interpret the results.

By
Ratheesan
24 REPLIES
Regular Contributor

## Re: Logistic Regression

Hi there,

There is a Wiki page on logistic regresssion, http://en.wikipedia.org/wiki/Logistic_regression, from which it looks like...

Intercept        =  β0 = a = Bias(offset) = 1.2
Coefficient      =  β1 = b = W[score1]  = 17.32
Coefficient      =  β2 = c = W[score2]  = 18.33

???
Regular Contributor

## Re: Logistic Regression

Thanks
Ratheesan
Regular Contributor

## Re: Logistic Regression

Hello,
I have one more doubt,ie,Is the model corresponding to probability of YES or probability of NO.

Thanks
Ratheesan
Regular Contributor

## Re: Logistic Regression

Greetings,

The regression enables you to work out the probabilities for each of the label values,  and to select the most probable, as you can see from the following...

`<?xml version="1.0" encoding="UTF-8" standalone="no"?><process version="5.0">  <context>    <input/>    <output/>    <macros/>  </context>  <operator activated="true" class="process" expanded="true" name="Root">    <description>Using a logistic regression learner for a classification task of numerical data.</description>    <process expanded="true" height="584" width="962">      <operator activated="true" class="retrieve" expanded="true" height="60" name="Retrieve" width="90" x="45" y="30">        <parameter key="repository_entry" value="../../data/Sonar"/>      </operator>      <operator activated="true" class="logistic_regression" expanded="true" height="94" name="MyKLRLearner" width="90" x="179" y="30">        <parameter key="calculate_weights" value="false"/>        <parameter key="return_optimization_performance" value="false"/>      </operator>      <operator activated="true" class="apply_model" expanded="true" height="76" name="Apply Model" width="90" x="431" y="28">        <list key="application_parameters"/>      </operator>      <connect from_op="Retrieve" from_port="output" to_op="MyKLRLearner" to_port="training set"/>      <connect from_op="MyKLRLearner" from_port="model" to_op="Apply Model" to_port="model"/>      <connect from_op="MyKLRLearner" from_port="exampleSet" to_op="Apply Model" to_port="unlabelled data"/>      <connect from_op="Apply Model" from_port="labelled data" to_port="result 2"/>      <connect from_op="Apply Model" from_port="model" to_port="result 1"/>      <portSpacing port="source_input 1" spacing="0"/>      <portSpacing port="sink_result 1" spacing="0"/>      <portSpacing port="sink_result 2" spacing="0"/>      <portSpacing port="sink_result 3" spacing="0"/>    </process>  </operator></process>`

Contributor II

## Re: Logistic Regression

I think my earlier post might be very similar to what ratheesan is wanting to know.

For a logistic regression problem, Rapid Miner calculates the intercept and the coefficients: β0,β1, and β2 in his example.

Then, according to http://en.wikipedia.org/wiki/Logistic_regression, we use these values to obtain a "z" value. For example,
z = β0 + β1*<attribute1 value> + β2*<attribute2 value>.

This "z" value is then evaluated in the equation 1/(1+e-z) in order to obtain the probabiliity that this particular instance will evaluate to "TRUE". Is this correct?

I am interested in extending a Rapid Miner model to evaluate real-time data and calculate confidence probabilities outside of RapidMiner and want to know if I am on the right track as to how to accomplish this. Unfortunately, when I follow the process outlined above, I am not obtaining the same confidence probabilities for data entries that are being calculated in RapidMiner.

Again, thanks for all the help!

David
Regular Contributor

## Re: Logistic Regression

Hi David,

 This "z" value is then evaluated in the equation 1/(1+e-z) in order to obtain the probabiliity that this particular instance will evaluate to "TRUE". Is this correct?

In a word, no. If you run the code I posted you will see that z is calculated for each possible label value, i.e the probability that this instance is a rock, and the probability that this instance is a mine.
Contributor II

## Re: Logistic Regression

Thanks for the reply! I am still confused, but I have high hopes I will understand this very soon.

Let me tell you where I am at ...
In your example you provided, I run it and view the Example Set in Data View. Here I can see each row, the class, the confidence(Rock), the confidence(Mine), the prediction(class), as well as each attribute value for each row.

Explicitly, how can I replicate the confidence values calculated for each row of data (is this the possible label value which you are speaking of)?

I have tried to replicate these values using the instructions discussed in the Logistic Regression wiki, but I do not get the same answers. Again, I would like to extend what I am discovering in RapidMiner to real-time confidence calculations of data.

Thanks again for all the education!

David
Regular Contributor

## Re: Logistic Regression

Is the confusion around the fitted parameters? Depending on if you are modeling 1 or 0, the signs of the coefficients flip....at least in other implementations I am familiar with (SAS).

I tried to look and see what RM was doing...but then I got worried.

I took the sonar data Haddock and used just used the first attribute as a predictor (see below)
The first example has attribute 1 = 0.020 so with the model returned here is what I would have expected:

`Bias	0.184weight_att 1	0.69	att1	0.02linear predictor (mu)	0.19781/(1+exp(-mu))	0.549289401`

But the two confidences RM gives are 0.6438660616438266 and 0.3561339383561734
Any ideas?

Here is the RM xml:
`<?xml version="1.0" encoding="UTF-8" standalone="no"?><process version="5.0">  <context>    <input>      <location/>    </input>    <output>      <location/>      <location/>      <location/>      <location/>      <location/>    </output>    <macros/>  </context>  <operator activated="true" class="process" expanded="true" name="Root">    <description>Using a logistic regression learner for a classification task of numerical data.</description>    <process expanded="true" height="298" width="614">      <operator activated="true" class="retrieve" expanded="true" height="60" name="Retrieve" width="90" x="45" y="30">        <parameter key="repository_entry" value="//Samples/data/Sonar"/>      </operator>      <operator activated="true" class="work_on_subset" expanded="true" height="94" name="Work on Subset" width="90" x="112" y="120">        <parameter key="attribute_filter_type" value="subset"/>        <parameter key="attributes" value="attribute_1|class"/>        <parameter key="include_special_attributes" value="true"/>        <parameter key="keep_subset_only" value="true"/>        <process expanded="true" height="298" width="614">          <connect from_port="exampleSet" to_port="example set"/>          <portSpacing port="source_exampleSet" spacing="0"/>          <portSpacing port="sink_example set" spacing="0"/>          <portSpacing port="sink_through 1" spacing="0"/>          <portSpacing port="sink_through 2" spacing="0"/>        </process>      </operator>      <operator activated="true" class="logistic_regression" expanded="true" height="94" name="Logistic Regression" width="90" x="246" y="75"/>      <operator activated="true" class="apply_model" expanded="true" height="76" name="Apply Model" width="90" x="447" y="165">        <list key="application_parameters"/>      </operator>      <connect from_op="Retrieve" from_port="output" to_op="Work on Subset" to_port="example set"/>      <connect from_op="Work on Subset" from_port="example set" to_op="Logistic Regression" to_port="training set"/>      <connect from_op="Work on Subset" from_port="through 1" to_port="result 4"/>      <connect from_op="Logistic Regression" from_port="model" to_op="Apply Model" to_port="model"/>      <connect from_op="Logistic Regression" from_port="weights" to_port="result 3"/>      <connect from_op="Logistic Regression" from_port="exampleSet" to_op="Apply Model" to_port="unlabelled data"/>      <connect from_op="Apply Model" from_port="labelled data" to_port="result 2"/>      <connect from_op="Apply Model" from_port="model" to_port="result 1"/>      <portSpacing port="source_input 1" spacing="0"/>      <portSpacing port="sink_result 1" spacing="0"/>      <portSpacing port="sink_result 2" spacing="0"/>      <portSpacing port="sink_result 3" spacing="0"/>      <portSpacing port="sink_result 4" spacing="0"/>      <portSpacing port="sink_result 5" spacing="0"/>    </process>  </operator></process>`

Regular Contributor

## Re: Logistic Regression

Hi,
I applied W-logistic instead of logistic regression on the above mentioned algorithm.There it clearly pointed that the model is corresponding to Class=ROCK.
So I think the result provided by Logistic Regression is also for class=ROCK. Is it right??

Thanks
Ratheesan