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# Fourier Series Question

1. May 8, 2005

### jwsiii

I'm having a hard time grasping exactly what a Fourier series is. I know the book definition and that it represents any periodic function as an infinite series. I also know that the a0 term is the average value of the function over one period. I can calculate the terms and everything but I don't really understand what I'm doing; I just know how to go through the calculations. Can someone give me a good, conceptual explanation of what a Fourier series is and how it works? Thanks.

2. May 8, 2005

### James R

A Fourier series expresses any function as a sum of sine and cosine waves of different frequencies. For example, if you're talking about sound, you can take a wave of any shape you like and extract the frequency components of that wave by expressing it as a Fourier series.

3. May 8, 2005

### Crosson

This my sound strange to you, but I think it may also be the best way to anser your question.

Presumambly you have worked with euclidean vectors: Arrows (points) in 3-space and 2-space. You may be familiar with an operation refered to as a scalar product, or dot product. If we take the product A dot B, we get the component of A in the direction of B.

It is possible to generalize generalize the idea of vectors to include functions, and to generalize the dot product as an integral. Essentially, what you are doing by taking the nth fourier integral is finding the component of f[x] in the Sin[n x] direction.

4. May 8, 2005

### jwsiii

Earlier in the math course I'm taking now, we did a block on vector calculus and learned the flux, which is the integral over a surface of the dot product of the vectors and the normal vector in a vector field. Is that similar to what's going on in a Fourier series in any way?

Also, I kind of understand the representation of a function as a vector but not quite. Is it like having a vector that points either straight up or down in the y-direction for each point on the x-axis? Thanks.

5. May 9, 2005

### mathwonk

A vector in the plane is an arrow that points from the origin to some other point (x,y).

it can be written as a "sum" of two simpler vectors, simpler because they are either horizontal or vertical. to find the terms in this sum, we ask of our vector (x,y) how much it points in the hporizontal direction, [answer (x,0)], and how much of it points in the vertical direction, [answer (0,y)].

one way to tell that a vector such as (0,6) does not point at all in the hporizontal direction is to note that it is perpendicular to the standard horizontal vector (1,0).

since dot products measure perpendicularity, they help decompose a vector into horizontal and vertical parts.

To say (0,6) is perpendicular to (1,0), means that as functions with only two values, they take on their values on different points in the domain. i.e. (0,6) is "supported" only on the second entry while (1,0) is supported on the first entry. so when we multiply their values: 0.1 + 6.0, we just get zero.

we also get zero when their values on these points cancel each other out, as in (2,1) and (-1,2), so the dot product is 2(-1) +(1(2).

to tell whether a function f is supported on the same or opposite points of its domain from another standard function like sin(nx) we also multiply them f(x)sin(nx), and then see how big this product is by integrating it.

for instance if f is large where sinj(nx) is small the integral should be small, and also if the values of f cancel out those of sin(nx).

Since vectors are just functions which have only two domain points, this idea of perpendicularity of functions is a direct generalization of that of vectors, and allows us to decide how much f points in the direction of sin(nx), and hence to write f as a sum of various standard functions sin(nx), cos(mx).

note we have to choose different standard functions here as functions like (1,0,...,0) with only one non zero entry are not detectable by integrating.

Actually sin and cos are not the best choices, better are e^nx and e^inx , since these standard functions are eigenfunctions for the operator D of differentiation.

of course sinx = (1/2i)[e^(x) - e^(ix)] and cos(x) = (1/2)(e^x + e^(ix)), so the distinction is not fatal.

Last edited: May 9, 2005
6. May 9, 2005

### fourier jr

there's a theorem that says: let $${x_1, x_2, ...}$$ be an orthonormal set in an inner product space V & let $$x \in V$$.
spose there exist scalars $$c_1, c_2, ... ,c_n$$ such that $$x = \sum_{k=1}^{n}c_k x_k$$ then $$c_k = <x,x_k>$$ for k=1,...,n

that sum thing is the Fourier series of the vector x. it's just a different way of writing it. that <x, x_k> bit is the inner product, which is just the dot product of 2 vectors. orthonormal set means you've got a bunch of unit vectors that are all perpendicular to each other.

the definition of Fourier series goes something like this:

let $$X = {x_1, x_2, ...}$$ be a countable orthonormal set in an inner product space V & let $$x \in V$$. the infinite series $$\sum_{n=1}^{\infty}<x, x_k>x_k$$ is called the Fourier series of x, & the coefficient <x, x_k> is called the Fourier coefficient of x.

that probably doesn't make much sense to a 2nd-yr or 3rd-yr student, but a countable orthonormal set could just be the standard basis for V=R^2: { (0,1), (1,0) } & the inner product would be the usual dot product. you can have different inner products & different "vectors" though; one inner product is the standard formula for Fourier coefficients: $$\frac{1}{L} \int_{L}^{L+2L}f(x)sin(\frac{n\pi x}{L})dx$$ (similarily for cosine series), & the "vectors" are sines & cosines, etc.

7. May 9, 2005

### mathwonk

that "theorem" is essentially distributivity of multiplication.


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# Extensive Labor Margin and Full Insurance

Extensive margin models use Rogerson (1988) style lotteries in order to simplify the savings problem under non-full-insurance.

However, one could still write down the model in that style but instead of assuming full insurance (that is, marginal utility of consumption equalized among all individuals in the household, no matter whether working or not), do it with partial insurance, in order to match the data better.

I however have not seen this approach so far. Did I miss something? Or what would be arguments against this approach? Besides the obvious one: If you don't have full insurance but forbid savings at the individual agent level, you are an evil person.

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What do you mean by partial insurance? Would self-insurance by borrowing and saving in an incomplete markets economy count? Or do you mean literally partial insurance, e.g. insured against some shocks but not others? The latter is in, for instance, Heathcote, Storesletten, and Violante's latest AER. –  nominally rigid Dec 18 '14 at 17:19
I mean that the unemployed individuals in the household get, say, 40% of the consumption that the employed ones get, to reflect the fact that in reality, unemployment insurance does not perfectly insure against unemployment –  FooBar Dec 18 '14 at 17:42
ah. If there's no borrowing/saving, and people who don't work get an unemployment benefit of 40% of the wage of people who do work, then not clear how employment would be affected by real wages: since return from employment and unemployment would move proportionally, for (e.g.) KPR preferences real wages would be neutral for employment. If, on the other hand, unemployment benefits are constant, then there's an unambiguous sign of employment response w.r.t. real wages, but magnitude still in question. –  nominally rigid Dec 19 '14 at 0:49
in fact, in benchmark Rogerson case with no heterogeneity, the elasticity of labor supply wrt real wages would still be either zero or infinity, probably zero - depending on whether you happened to be at the point where everyone was indifferent between working and not working. Without agents being able to borrow and save, the model behaves in strange ways, which is why ivansml's focus below on self-insurance through borrowing/saving makes sense. –  nominally rigid Dec 19 '14 at 0:52

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# Nonparametric adaptive estimation for integrated diffusions.

Abstract : We consider here nonparametric estimation for integrated diffusion processes. Let $(V_t)$ be a stationary and $\beta$-mixing diffusion with unknown drift and diffusion coefficient. The integrated process $X_t= \int_0^{t} V_s ds$ is observed at discrete times with regular sampling interval $\Delta$. For both the drift function and the diffusion coefficient of the unobserved diffusion $(V_t)$, we propose nonparametric estimators based on a penalized least square approach. Estimators are chosen among a collection of functions belonging to a finite dimensional space selected by an automatic data-driven method. We derive non asymptotic risk bounds for the estimators. Interpreting these bounds through the asymptotic framework of high frequency data, we show that our estimators reach the minimax optimal rates of convergence. The algorithms of estimation are implemented for several examples of diffusion models that can be exactly simulated.
keyword :
Type de document :
Article dans une revue
Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.811-834. <10.1016/j.spa.2008.04.009>
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https://hal.archives-ouvertes.fr/hal-00110510
Contributeur : Fabienne Comte <>
Soumis le : lundi 30 octobre 2006 - 14:11:16
Dernière modification le : mardi 11 octobre 2016 - 12:01:01
Document(s) archivé(s) le : mardi 6 avril 2010 - 21:17:17

### Citation

Fabienne Comte, Valentine Genon-Catalot, Yves Rozenholc. Nonparametric adaptive estimation for integrated diffusions.. Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.811-834. <10.1016/j.spa.2008.04.009>. <hal-00110510>

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The distribution of H I--deficient galaxies isconcentrated at around 14--20 Mpc, indicating the presence of a core atthis distance, and this agrees with the distance estimated for M87 andother elliptical galaxies with other methods. We show also that thespatial number density of spiral galaxies takes a peak at this distance,while a simple average of all spiral galaxy distances gives 20 Mpc. Thefact that the velocity dispersion of galaxies takes a maximum at 14--18Mpc lends an additional support for the distance to the core. Thesefeatures cannot be understood if the large scatter of the TF relation ismerely due to the intrinsic dispersion. The structure of the VirgoCluster we infer from the Tully-Fisher analysis looks like a filamentwhich is familiar to us in a late phase of structure formation in thepancake collapse in hierarchical clustering simulations. This Virgofilament lies almost along the line of sight, and this is the originthat has led a number of authors to much confusion in the Virgo distancedeterminations. We show that the M87 subcluster is located around 15--18Mpc, and it consists mainly of early-type type spiral galaxies inaddition to elliptical and S0 galaxies. There are very few late-typespiral galaxies in this subcluster. The spiral rich M49 subclusterconsists of a mixture of all types of spiral galaxies and is located atabout 22 Mpc. The two other known clouds, W and M, are located at about30--40 Mpc and undergo infall toward the core. The M cloud contains fewearly type spirals. We cannot discriminate, however, whether thesesubclusters or clouds are isolated aggregates or merely parts offilamentary structure. Finally, we infer the Hubble constant to be 82+/- 10 km s-1 Mpc-1. Galaxies with a UV excess in which supernovae have been observed.Not Available A Minnesota Automated Plate Scanner Catalog of Galaxies behind the Virgo Cluster and toward Its AntipodeWe present a catalog of 1268 galaxies, essentially complete to B <=17.0, found by scanning glass copies of several fields of the originalPalomar Sky Survey using the Minnesota Automated Plate Scanner in itsisodensitometric mode (as opposed to the threshold densitometric modeused in the APS Catalog of the POSS I). In addition to the differentscanning mode, we have employed a different star-galaxy separationmethod and have visually inspected POSS prints to verify that each imageremaining in the catalog is nonstellar. The scanned fields aredistributed generally in two areas, one around the outskirts of theVirgo Cluster, the other toward the antipode of the cluster (but stillin the northern celestial hemisphere). The catalog gives the position ofthe center of each galaxy; estimates of the blue and red magnitudeswithin the outermost threshold crossing and of the blue magnitudeextrapolated to zero surface brightness; and the blue and red diametersof four ellipses fitted to the four threshold crossings (approximately23.8,23.6,23.2, and 22.7 mag arcsec 2 in blue, and 22.5,22.4,21.5, and21.2 mag arcsec^-2^ in red), and the ellipticities of those fourellipses. The catalog has served as a base from which to draw targetsfor a Tully-Fisher study of the Virgocentric infall velocity of theLocal Group. Explosion Models for Type IA Supernovae: A Comparison with Observed Light Curves, Distances, H 0, and Q 0Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1996ApJ...457..500H&db_key=AST Surface photometry of spiral galaxies in the Virgo cluster regionPhotographic surface photometry is carried out for 246 spiral galaxiesin the Virgo cluster region north of declination + 5 deg. The samplecontains all spiral galaxies of 'certain' and 'possible' Virgo membersin the Virgo Cluster Catalogue of Binggeli, Sandage, & Tammann. Thesample also includes those galaxies which were used in the Tully-Fisheranalyses of the Virgo cluster given in the literature. A catalog ispresented for positions, B-band total magnitudes and inclinations forthese galaxies, and they are compared with the data given in previousstudies. The Hubble Diagram for Supernovae of Type Ia. II. The Effect on the Hubble Constant of a Correlation between Absolute Magnitude and Light Decay RateNew Hubble diagrams in B and V are derived for supernovae of type Ibased on light curves from the archive literature plus 13 new lightcurves with superior modern photometry observed in the CerroTololo/University of Chile program (Hamuy et al, 1995). The sample isrestricted to SNe Ia whose light curves are defined by photometrybeginning 5 days or less after maximum light and with (B - V)max <0.5 mag. Supernovae of known type Ib or Ic are also excluded. Theresulting Hubble diagrams, extending to redshifts of 30,00 km s^- 1^,have dispersions in absolute magnitude of 0.34 mag in B and 0.33 mag inV, confirming that spectroscopically "normal" (Branch et al. 1993) SNeIa are among the best standard candles known. A solution for the slopeof the Hubble diagram gives n(B) = 0.977 +/- 0.025 and n(V) = 1.020 +/-0.024 for the exponent in ν~D^n^, proving linearity of the expansionfield to a high level. The residuals in magnitude from the ridge line ofthe Hubble diagram are compared with the light decay rate during thefirst 15 days to test the correlation between the two suggested byPskovskii and by Phillips. The strongest possible correlation using theextant data has a slope 3 times smaller than that derived by Phillips,and 2 times smaller than suggested by Hamuy et al., leading to adecrease of less than 10% in the distance scale based on the present(1995) SNe Ia calibration by means of three supernovae whose distancesare known from Cepheids in their parent galaxies. Applying the maximumpossible correction to M(max) for a Psko'vskii- Phillips effect wouldgive Hubble constants of H_0_(B)<= 54 +/- 4 km s^-1^ Mpc^-1^, andH_0_(V) <= 59 +/- 4 km s^-1^ Mpc^-1^, where the errors are internal.It is argued that the absence of measurable bias effects in the Hubblediagrams shows that the three local (nearest) SNe Ia presentlycalibrated via Cepheid distances cannot all be overluminous relative tothe average of more distant SNe. If they are underluminous, which mustbe the case by the statistics of the Malmquist effect if the largedispersion in M(max) for SNe Ia claimed by Hamuy et al. applies to thecalibrators, then the value of H_0_ = 52 km s^-1^ Mpc^-1^ given by Sahaet al. is an upper limit to the Hubble constant. The UV properties of normal galaxies. III. Standard luminosity profiles and total magnitudes.In the previous papers of this series we collected and reduced to thesame system all the available photometric data obtained in theultraviolet (UV) range for normal (i.e. non active) galaxies. Here weuse these data to derive standard UV luminosity profiles for threemorphological bins (E/S0; Sa/Sb; Sc/Sd) and extrapolated totalmagnitudes for almost 400 galaxies. We find that: 1) the UV growthcurves are well matched by the B-band revised standard luminosityprofiles, once a proper shift in the effective radius is applied, and 2)the UV light in early-type galaxies is more centrally concentrated thanthe visible light. The UV properties of normal galaxies. II. The non-IUE'' data.In the last decade several satellite and balloon borne experiments havecollected a large number of ultraviolet fluxes of normal galaxiesmeasured through apertures of various sizes and shapes. We havehomogenized this data set by deriving scale corrections with respect toIUE. In a forthcoming paper these data will be used to derive standardluminosity profiles and total magnitudes. Distribution of supernovae relative to spiral arms and H II regionsWe have studied the association of supernovae in spiral galaxies withsites of recent stars formation -- sprial arms and H II regions. It isshown that supernovae (SNe) of Types Ia, Ib, and II exhibitconcentration to spiral arms and their distributions over the distanceto the nearest spiral arm do not differ significantly. This result isconfirmed by a Kolmogorov-Smirnov test comparison with the distancedistributions, expected if SNe are distributed randomly inside the modelgalaxy. SNe of types Ib and II show a strong concentration towards H IIregions, while distribution of SNe Ia can be explained by chancesuperposition. All studied distributions of SNe Ib and II show strikingsimilarity, which suggests that their progenitors are massive stars withsimilar ages and initial masses. The association of SNe Ia with spiralarms suggests that their progenitors in spiral galaxies are likely to beintermediate mass stars. Arm structure in normal spiral galaxies, 1: Multivariate data for 492 galaxiesMultivariate data have been collected as part of an effort to develop anew classification system for spiral galaxies, one which is notnecessarily based on subjective morphological properties. A sample of492 moderately bright northern Sa and Sc spirals was chosen for futurestatistical analysis. New observations were made at 20 and 21 cm; thelatter data are described in detail here. Infrared Astronomy Satellite(IRAS) fluxes were obtained from archival data. Finally, new estimatesof arm pattern radomness and of local environmental harshness werecompiled for most sample objects. A comparison of calculated and observed monochromatic Type IA supernova light curvesThe calculated monochromatic (B and V) light curves of a set of Type Iasupernova models are compared with the observed monochromatic lightcurves of 13 Type Ia supernovae (SNIa). The set consists of 11 Type Iasupernova models including deflagrations, detonations, delayeddetonations, pulsating delayed detonations and tamped detonations. Thetheoretical light curves have been calculated using our recentlydeveloped light curve model. The model incorporates a detailed equationof state with an elaborate treatment of the ionization balance, animplicit grey Local Thermodynamic Equilibrium (LTE) radiation transportscheme based on the moment equations, a total energy equation for thegas and the radiation field, expansion opacities, and a Monte Carlogamma-ray deposition scheme. To evolve the structure of the underlyingexplosion models homologous expansion is assumed. We find that fastrising light curves (e.g., SN 1972E, SN 1981B and SN 1986G) can bereproduced best by delayed detonation models, and that slowly rising(tmax greater than or approximately equal to 17 days) lightcurves (e.g., SN 1984A and SN 1990N) cannot be reproduced by standarddetonation, deflagration or delayed detonation models. In order toobtain an acceptable agreement with observations for slowly rising lightcurves explosion models are required where the C/O white dwarf issurrounded by an unburnt extended envelope of typically 0.2 to 0.4 solarmass. Our interpretation of the light curves is also favored from theexpansion velocities observed in the spectra of the slowly rising SNIa.Based on a comparison of theoretical light curves and observationaldata, the distances of the parent galaxies have been determined. We showthat SNeIa allow for a measurement of the value of the Hubble constantprovided individual fits are used and if they are located at distanceslarger than 15 Mpc, because only then they are in the 'Hubble' flow.Taking this restriction into account a Hubble constant of H = 66 +/- 10km/(secMpc) can be derived within a 2 sigma error. The Hubble diagram in V for supernovae of Type IA and the value of H(0) therefromThe Hubble diagram for Type I supernovae is derived in V and issummarized from the literature in B and in m(pg). The ridge lineequation of the diagram in V and the calibration of the absolutemagnitudes at maximum are presented. The intrinsic (B - V) color at Bmaximum light is 0.09 +/- 0.04 mag. The Virgo Cluster distance isderived and found to be 23.9 +/- 2.4 Mpc. This Virgo distance gives thecosmic value of the Hubble constant to be H(0) = 47 +/- 5 km/sec perMpc. The kinematics of the Virgo cluster revisitedThe paper updates the velocity data of Virgo cluster galaxies andreconsiders the kinematic structure of the Virgo cluster. New velocitiesare given for 144 galaxies listed in the Virgo Cluster Catalog (VCC).Improved velocities are given for another 131 VCC galaxies. The Virgocluster is disentangled from its surrounding clouds of galaxies, and thelikely members of each of these clouds are listed. The velocitydistribution of dwarf elliptical cluster members is found to be highlyasymmetric. This phenomenon is interpreted as evidence for the imminentmerging of two subclusters in the core region, which points to thedynamical youth of the Virgo cluster. The mean heliocentric velocity ofthe Virgo cluster is estimated at 1050 +/- 35 km/s.
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#### Observation and Astrometry data

 Constellation: Virgo Right ascension: 12h12m11.80s Declination: +13Â°14'47.0" Aparent dimensions: 1.23′ × 0.708′

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# Learning Machines

### Week 5

#### Unsupervised Learning

In a supervised learning procedure, the machine tries to learn the underlying relationships between a set of example inputs and their corresponding outputs.

In an unsupervised learning procedure, the machine only tries to learn about the internal structure of the input examples.

This allows us to extract meaningful features and patterns from the input data.

Unsupervised learning is useful in various forms of data analysis and dimensionality reduction.

Unsupervised learning can also be applied to generative or creative applications. For example, we could learn about the kinds of patterns that exist within a particular composer's work and then use this to generate new compositions in the style of that artist.

It can also be useful in improving the quality of a supervised learning procedure. This is particularly true in cases where we have far more input examples than we have correctly labeled corresponding outputs. For example, the internet provides an endless supply of images from which we could learn internal patterns. But only a small portion of these images come with reliable descriptions of the images' contents. Unsupervised learning allows us to learn common patterns across both labeled and unlabeled images and then use these underlying patterns to associate previously unlabeled images with labels that correspond to structurally similar images.

This approach bears some resemblance to our own learning process in the sense that we have many experiences interacting with a particular kind of object, but a much smaller number of experiences in which another person explicitly tells us the name of that object.

Unsupervised learning in general and in its relation to supervised learning can tell us a lot about the nature of thought and the representation of ideas.

#### Restricted Boltzmann Machine Architecture

##### Conceptual Overview

Based on my article Machine Learning and Improvisation.

A Restricted Boltzmann Machine (RBM) is a two-layer neural network composed of a visible layer and a hidden layer.

Each unit in the visible layer is connected to each unit in the hidden layer. The visible units are not connected to one another and the hidden units are not connected to one another.

An RBM observes real-world patterns and tries to create a lower-dimensional representation of those patterns.

We can also stack multiple RBMs on top of one another to reduce the dimensionality of the patterns even further. One common architecture for stacked RBMs is called a Deep Belief Network (DBN).

The goal of the training process is to produce lower dimensional representations that can then be used to “reconstruct” approximations of the original inputs.

In the animation below, we will train a DBN on leaf images from 100 species, using 16 images per species.

You can think of this as a kind of compression algorithm, improvised in relation to the neural network’s experience.

The network compresses information by finding component patterns across many input examples. It can then use these component patterns as building blocks to describe the whole.

This is an efficient way to store information because it means we don’t need to hold onto every detail of every input example. We can use a more general vocabulary derived from all of the examples to describe each particular example.

The reconstruction of an input example through this process is not exact. But that’s what’s interesting about it!

Notice how the approximation changes as we go to deeper layers of the neural network:

And notice what happens when we reconstruct partially obfuscated images:

This is somewhat like our minds filling in the missing pieces of a face that has been partially occluded by some other object such as a telephone pole.

Representing experiences through a shared set of component patterns means that we don’t have to treat each as entirely separate from or incomparable to each other. We can fill gaps in our experiences by borrowing from our memory of other similar experiences.

##### Tracing the Path of a Single Input

The $$x$$ component is multiplied by a $$weight$$ and added to a $$bias$$. The result of this operation is then passed through an activation function.

The resulting $$activation$$ represents the strength of the signal passing through the hidden node, given input $$x$$.

More generally:

$$activation = activationFunction( weight * input + hiddenBias )$$

##### Combining Weighted Inputs

Each $$x$$ component is multiplied by a separate $$weight$$. These products are summed together and added to a $$bias$$. The result of this operation is passed through an activation function.

##### Multiple Inputs

Each hidden node receives the components of the input signal, multiplied by their corresponding $$weights$$. These products are summed together and added to the hidden node's $$bias$$. The results of these operations are passed through an activation function.

##### Reconstruction

During reconstruction, the hidden activations are passed backward through the network.

The $$activations$$ are multiplied by their corresponding $$weights$$. These products are summed together and added to the visible node's $$bias$$. The results of these operations are passed through an activation function.

More generally:

$$reconstruction = activationFunction( {activation}^{T} * weight + visibleBias )$$

The output of this reconstruction process acts as an approximation of the original input.

##### Abstraction and Deabstraction

The forward and backward passes described above give us two symmetrical processes for abstracting and deabstracting (or, compressing and decompressing) information:

Get Hidden Activations (Propagate Up)

$$activationsH = \frac{1}{1 + {e}^{-(inputV \cdot weights + biasH)}} \\$$

Get Visible Activations (Propagate Down)

$$activationsV = \frac{1}{1 + {e}^{-(inputH \cdot {weights}^{T} + biasV)}} \\$$

#### Training Restricted Boltzmann Machines

Based on CSC321: Neural Networks, by Geoffrey Hinton.

##### An Inefficient Approach

Learning Rule

$$\Delta{{W}_{ij}} = learningRate * ( {\langle{v}_{i}{h}_{j}\rangle}^{0} - {\langle{v}_{i}{h}_{j}\rangle}^{\infty} )$$

Learning Procedure

• Apply a training example to the visible units
• Alternate between:
• Updating all hidden units in parallel
• Updating all visible units in parallel
• After many alternations (timestep $$\infty$$), the system will reach thermal equilibrium.
• We can now measure the correlation between $$v_i$$ and $$h_j$$.

Limitations

• We have to perform this alternation many times before it reaches thermal equilibrium.
• If we don't perform it a sufficient number of times, the learning may go wrong.
• But, it turns out that we can take a shortcut: Contrastive Divergence.
##### Contrastive Divergence

Learning Rule

$$\Delta{{W}_{ij}} = learningRate * ( {\langle{v}_{i}{h}_{j}\rangle}^{0} - {\langle{v}_{i}{h}_{j}\rangle}^{1} )$$

Learning Procedure

• Apply a training example to the visible units
• Update all hidden units in parallel
• Update all visible units in parallel
• Update all hidden units again

Instead of waiting for thermal equilibrium, we use the measurements produced by one update step.

Why Contrastive Divergence Works

• If we start from the data, the Markov chain will wander away from the data values themselves and towards values it likes more (an equilibrium distribution).
• After only a few steps, we can see what direction it is wandering in.
• Since we know the initial weights are bad, it is wasteful to let them go all the way to equilibrium. We know how to change the weights to stop them from wandering away from the data without going all the way to equilibrium.
• We simply need to lower the probability of the reconstructions it produces after one full step and then raise the probability of the data.
• This will stop the chain from wandering away from the data.
• The learning will cancel out once the reconstructions and the data have the same distribution as one another.

#### Implementing a Restricted Boltzmann Machine

import numpy as np

def sigmoid(x):
'''sigmoid function'''
return 1.0 / ( 1.0 + np.exp( -x ) )

class Rbm:
def __init__(self, name, sizeV, sizeH, continuous = False):
self.name    = name
self.is_crbm = continuous
# Initialize weights:
self.weights = np.array( np.random.uniform( -1.0 / sizeV, 1.0 / sizeV, ( sizeV, sizeH ) ) )
# Initialize biases:
self.biasH   = np.zeros( sizeH )
self.biasV   = np.zeros( sizeV )

def getErrorRate(self, samples, reconstructions):
'''returns mean square error'''
return np.mean( np.square( samples - reconstructions ) )

def trainEpoch(self, training_samples, learn_rate, cd_steps, batch_size):
error      = 0.0
num_rows   = training_samples.shape[ 0 ]
# Iterate over each training batch:
for bstart in range( 0, num_rows, batch_size ):
# Compute batch stop index:
bstop = min( bstart + batch_size, num_rows )
# Compute batch size:
bsize = bstop - bstart
# Compute batch multiplier:
bmult = learn_rate * ( 1.0 / float( bsize ) )
# Slice data:
bsamples = training_samples[ bstart:bstop, : ]
# Get hidden activations and samples:
aH_0, sH_0 = self.getHiddenSample( bsamples )
# Perform each contrastive divergence step:
for i in range( cd_steps ):
aV_inf, sV_inf, aH_inf, sH_inf = self.getGibbsHvh( ( sH_0 if i == 0 else sH_inf ) )
# Update weights:
self.weights += bmult * ( np.dot( bsamples.T, aH_0 ) - np.dot( sV_inf.T, aH_inf ) )
# Update biases:
self.biasV   += bmult * np.mean( bsamples - sV_inf, axis = 0 )
self.biasH   += bmult * np.mean( aH_0 - aH_inf, axis = 0 )
# Scale batch error and accumulate total:
error += self.getErrorRate( bsamples, sV_inf ) * ( float( bsize ) / float( num_rows ) )
# Return training error:
return error

def train(self, training_samples, validation_samples, learn_rate, cd_steps, epochs, batch_size = 10, report_freq = 10, report_buff = 100):
# Perform each training epoch:
for epoch in range( epochs ):
training_error = self.trainEpoch( training_samples, learn_rate, cd_steps, batch_size )
if ( epoch + 1 ) % report_freq == 0:
# Compute validation error:
val_aH, val_sH, val_aV, val_sV = self.getGibbsVhv( validation_samples )
validation_error = self.getErrorRate( validation_samples, val_sV )

def getHiddenActivations(self, inputV):
return sigmoid( np.dot( inputV, self.weights ) + self.biasH )

def getVisibleActivations(self, inputH):
return sigmoid( np.dot( inputH, self.weights.T ) + self.biasV )

def getHiddenSample(self, inputV, force_binomial = False):
aH = self.getHiddenActivations( inputV )
return [ aH, self.getSample( aH ) if ( force_binomial or not self.is_crbm ) else aH ]

def getVisibleSample(self, inputH, force_binomial = False):
aV = self.getVisibleActivations( inputH )
return [ aV, self.getSample( aV ) if ( force_binomial or not self.is_crbm ) else aV ]

def getGibbsHvh(self, inputH, force_binomial = False):
aV, sV = self.getVisibleSample( inputH, force_binomial )
aH, sH = self.getHiddenSample( sV, force_binomial )
return [ aV, sV, aH, sH ]

def getGibbsVhv(self, inputV, force_binomial = False):
aH, sH = self.getHiddenSample( inputV, force_binomial )
aV, sV = self.getVisibleSample( sH, force_binomial )
return [ aH, sH, aV, sV ]

@staticmethod
def getSample(activations):
return np.random.binomial( 1, activations, activations.shape )

Take a look at the full Restricted Boltzmann Machine example code.

#### Getting Started with Docker and LaunchBot

Next week, we will be using Google’s TensorFlow library.

To prepare for that, I would like to spend a bit of time at the end of class installing Docker and LaunchBot, two tools that will help us to streamline our TensorFlow work.

Please see the LaunchBot Getting Started Tutorial for instructions on how to install Docker and LaunchBot.

Assignment


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# 2D Ray Tracing

Last year I experimented with using a 2D ray tracer as a debugging tool when working with new sampling techniques. Running the ray tracer in two dimensions often helped to simplify problems and made it easier to visualize the processes involved. The tool had two modes of operation — the first was a 2D path tracer that emitted rays from a sensor and the second emitted directly from light sources. The latter mode, when I referred to as “light tracing”, was mainly a gimmick. It was occasionally helpful for debugging ray–shape intersection code, though, and produce cool visual results.

The light tracing mode

The light tracing performance was decent, especially after parallelizing it, so I ended up trying to make it run at interactive framerates. While the resulting demo was fun to play with, the convergence time per frame was still too long for any practical uses.

# Enter WebGL

Last weekend I decided to try implementing a similar effect in WebGL. Rather than shading the entire scene with ray tracing code, this time I only used it to find shadowed/occluded regions and darken them accordingly. The rest of the shading was done using standard rasterization techniques. Both the performance and visual quality improved significantly with the WebGL approach. The image below shows the scene I used in my demo; the demo itself can be found in my WebGL portfolio. Moving the mouse changes the location of one of the lights in the scene and clicking anywhere in the canvas will toggle a Gaussian blur effect.

WebGL light tracing, computed per-pixel

# Implementation Details

Since it’s written in WebGL, the source code is available on the demo page itself. Admittedly, some portions are a bit convoluted and I haven’t really cleaned it up.

The entire scene is rendered with a single full screen quad. One render pass is made over the quad and a final color value for each pixel is computed based on the scene state. This process produces the lighting and shadows, circular occluders, and the background grid. The Gaussian blur effect is done in two subsequent

The shader starts off with a base color for the active pixel. Modular arithmetic is then used to determine if the pixel is a grid line and the pixel brightness is increased accordingly. Next, each light in the scene is considered. The distance from the pixel to the light is found and an inverse falloff model is used to compute the amount of light that reaches the pixel. Additionally, a darkening factor is computed for the light by comparing it against the occluders in the scene. This is the most interesting portion and the part that is largely based on ray tracing.

To compute the shadows a light ray is traced from each light source to the active pixel. Ray tests are performed on the circles in the scene to determine if the light ray actually reaches the active pixel or if it hits the circle. The result of these collision tests determine if the pixel is shadowed or not. If a pixel is shadowed is darkening factor is assumed to be 0.0, otherwise it’s 1.0.

Since the light sources are treated as points, soft shadowing isn’t present by default. To achieve a smoother edge and fade out of the shadows I introduced a few fudge factors into various portions of the code. These were determined empirically — they have no physical basis whatsoever.

First, the circle–ray collision code was modified to return a value on the range [0.0, 1.0] instead of a simple shadowed/lit boolean. A value of 0.0 still indicated a completely shadowed pixel, but values between 0.0 and 1.0 were used to describe partial/blurry shadowing. When a ray–circle test is performed, one of the intermediate steps is to compute a discriminant when solving the quadratic equation for the intersection. The value of the discriminant tends to be quite small when the ray is approximately tangent to the circle and increases as the ray moves closer to the circle’s center — this property was used to produce the soft shadow approximation. Additionally, the distance from the ray to the light was used to make the the shadow fade out with increased distance. The final formula for the fudge factor was:

$$factor=\frac{d*0.035}{\sqrt{D}}$$

where d is the distance from the pixel to the light and D is the discriminant. The scale factor of 0.035 was determined by tweaking the value until a suitable shadow brightness was achieved. The result of the factor expression was added to the original 0.0 or 1.0 value from the ray test and then clamped to the range [0.0 1.0]. Pixels with a small value of D or a large value of d evaluated to large factor values, thus lightening the pixel.

The final step in the rendering process was to shade the gray circles. This was done last so the tops of the circles wouldn’t be shadowed. Circles were filled by simply checking if the active pixel was inside one of the circles and shading that pixel accordingly.

# Performance and Practicality

The demo runs quite nicely on all the machines I’ve tested it on, however there are definitely some areas that need optimization. For larger numbers of occluders a quad tree acceleration structure could be used to speed up the ray tracing step. Right now the demo only works with point lights and circular occluders — ray testing against arbitrary shapes should be possible, but requires additional collision routines. That said, the technique isn’t really necessary in most cases. A similar effect can be replicated using a 2D variant of shadow mapping, which also scales better for larger scenes and more complex objects.

One thing I’d like to do is try the faked soft shadowing approach used here in a traditional single-bounce ray tracer. I’m curious how the results compare to a path tracer with true soft shadowing and area light support. In terms of performance the faked shadow version would be significantly faster, at the expense of global illumination quality. If the results are sufficiently convincing, the faked shadows could be used for early test renders before committing to a long render time in a proper path tracer.


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# Chapter 13 - Probability - Mid-Chapter Quiz - Page 843: 2

$\frac{9}{39}$

#### Work Step by Step

To calculate probability, we need to divide the number of chances for a certain outcome to occur by the total number of outcomes. In this case, there are 9 tiles that are green, and 39 in total. When we divide 9 by 39, we get $\frac{9}{39}$, which is the probability of drawing a green tile, or $P(\text{green})$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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location:  Publications → journals
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26. CJM 1997 (vol 49 pp. 1265)

Snaith, V. P.
 Hecke algebras and class-group invariant Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide. Categories:16S34, 19A99, 11R65
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# Polynomial-time approximation scheme

Last updated

In computer science, a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems).

Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.

In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to NP-hard optimization problems with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for Scheduling on Unrelated Parallel Machines.

In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete variables is known as a discrete optimization. In a discrete optimization problem, we are looking for an object such as an integer, permutation or graph from a countable set. Problems with continuous variables include constrained problems and multimodal problems.

## Contents

A PTAS is an algorithm which takes an instance of an optimization problem and a parameter ε > 0 and, in polynomial time, produces a solution that is within a factor 1 + ε of being optimal (or 1 ε for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most (1 + ε)L, with L being the length of the shortest tour. [1] There exists also PTAS for the class of all dense constraint satisfaction problems (CSPs). [2] [ clarification needed ]

The running time of a PTAS is required to be polynomial in n for every fixed ε but can be different for different ε. Thus an algorithm running in time O (n1/ε) or even O(nexp(1/ε)) counts as a PTAS.

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation.

## Variants

### Deterministic

A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is O(n(1/ε)!). One way of addressing this is to define the efficient polynomial-time approximation scheme or EPTAS, in which the running time is required to be O(nc) for a constant c independent of ε. This ensures that an increase in problem size has the same relative effect on runtime regardless of what ε is being used; however, the constant under the big-O can still depend on ε arbitrarily. Even more restrictive, and useful in practice, is the fully polynomial-time approximation scheme or FPTAS, which requires the algorithm to be polynomial in both the problem size n and 1/ε. All problems in FPTAS are fixed-parameter tractable. Both the knapsack problem and bin packing problem admit an FPTAS. [3]

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.

In the bin packing problem, objects of different volumes must be packed into a finite number of bins or containers each of volume V in a way that minimizes the number of bins used. In computational complexity theory, it is a combinatorial NP-hard problem. The decision problem is NP-complete.

Any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have an FPTAS unless P=NP. [4] However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded. [5]

Unless P = NP, it holds that FPTAS  PTAS   APX. [6] Consequently, under this assumption, APX-hard problems do not have PTASs.

In complexity theory the class APX is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant. In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer.

Another deterministic variant of the PTAS is the quasi-polynomial-time approximation scheme or QPTAS. A QPTAS has time complexity ${\displaystyle n^{\operatorname {polylog} (n)}}$ for each fixed ${\displaystyle \varepsilon >0}$.

### Randomized

Some problems which do not have a PTAS may admit a randomized algorithm with similar properties, a polynomial-time randomized approximation scheme or PRAS. A PRAS is an algorithm which takes an instance of an optimization or counting problem and a parameter ε > 0 and, in polynomial time, produces a solution that has a high probability of being within a factor ε of optimal. Conventionally, "high probability" means probability greater than 3/4, though as with most probabilistic complexity classes the definition is robust to variations in this exact value (the bare minimum requirement is generally greater than 1/2). Like a PTAS, a PRAS must have running time polynomial in n, but not necessarily in ε; with further restrictions on the running time in ε, one can define an efficient polynomial-time randomized approximation scheme or EPRAS similar to the EPTAS, and a fully polynomial-time randomized approximation scheme or FPRAS similar to the FPTAS. [4]

## As a complexity class

The term PTAS may also be used to refer to the class of optimization problems that have a PTAS. PTAS is a subset of APX, and unless P = NP, it is a strict subset. [6]

Membership in PTAS can be shown using a PTAS reduction, L-reduction, or P-reduction, all of which preserve PTAS membership, and these may also be used to demonstrate PTAS-completeness. On the other hand, showing non-membership in PTAS (namely, the nonexistence of a PTAS), may be done by showing that the problem is APX-hard, after which the existence of a PTAS would show P = NP. APX-hardness is commonly shown via PTAS reduction or AP-reduction.

## Related Research Articles

The ♯P-complete problems form a complexity class in computational complexity theory. The problems in this complexity class are defined by having the following two properties:

NP-hardness, in computational complexity theory, is the defining property of a class of problems that are, informally, "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem.

In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to differ by at most a constant factor.

In Operations Research, applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems, in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Some common problems involving combinatorial optimization are the travelling salesman problem ("TSP") and the minimum spanning tree problem ("MST").

In graph theory, a dominating set for a graph G = (VE) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G.

In graph theory, a directed graph may contain directed cycles, a one-way loop of edges. In some applications, such cycles are undesirable, and we wish to eliminate them and obtain a directed acyclic graph (DAG). One way to do this is simply to drop edges from the graph to break the cycles. A feedback arc set (FAS) or feedback edge set is a set of edges which, when removed from the graph, leave a DAG. Put another way, it's a set containing at least one edge of every cycle in the graph.

The Valiant–Vazirani theorem is a theorem in computational complexity theory stating that if there is a polynomial time algorithm for Unambiguous-SAT, then NP = RP. It was proven by Leslie Valiant and Vijay Vazirani in their paper titled NP is as easy as detecting unique solutions published in 1986. The proof is based on the Mulmuley–Vazirani–Vazirani isolation lemma, which was subsequently used for a number of important applications in theoretical computer science.

MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a decision problem considered in complexity theory. It is defined as:

In computational complexity theory, the maximum satisfiability problem (MAX-SAT) is the problem of determining the maximum number of clauses, of a given Boolean formula in conjunctive normal form, that can be made true by an assignment of truth values to the variables of the formula. It is a generalization of the Boolean satisfiability problem, which asks whether there exists a truth assignment that makes all clauses true.

In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters.

In computational complexity theory, a PTAS reduction is an approximation-preserving reduction that is often used to perform reductions between solutions to optimization problems. It preserves the property that a problem has a polynomial time approximation scheme (PTAS) and is used to define completeness for certain classes of optimization problems such as APX. Notationally, if there is a PTAS reduction from a problem A to a problem B, we write .

In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource used by the algorithm, averaged over all possible inputs. It is frequently contrasted with worst-case complexity which considers the maximal complexity of the algorithm over all possible inputs.

For a graph, a maximum cut is a cut whose size is at least the size of any other cut. The problem of finding a maximum cut in a graph is known as the Max-Cut Problem.

In computational complexity theory, a problem is NP-complete when it can be solved by a restricted class of brute force search algorithms and it can be used to simulate any other problem with a similar algorithm. More precisely, each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested quickly, such that the output for any input is "yes" if the solution set is non-empty and "no" if it is empty. The complexity class of problems of this form is called NP, an abbreviation for "nondeterministic polynomial time". A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it, and a problem is NP-complete if it is both in NP and NP-hard. The NP-complete problems represent the hardest problems in NP. If any NP-complete problem has a polynomial time algorithm, all problems in NP do. The set of NP-complete problems is often denoted by NP-C or NPC.

In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.

In computability theory and computational complexity theory, especially the study of approximation algorithms, an approximation-preserving reduction is an algorithm for transforming one optimization problem into another problem, such that the distance of solutions from optimal is preserved to some degree. Approximation-preserving reductions are a subset of more general reductions in complexity theory; the difference is that approximation-preserving reductions usually make statements on approximation problems or optimization problems, as opposed to decision problems.

In computer science and operations research, exact algorithms are algorithms that always solve an optimization problem to optimality. Unless P = NP, such an algorithm cannot run in worst-case polynomial time but there has been extensive research on finding exact algorithms whose running time is exponential with a low base.

## References

1. Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998.
2. Arora, S.; Karger, D.; Karpinski, M. (1999), "Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems", Journal of Computer and System Sciences, 58 (1): 193–210, doi:10.1006/jcss.1998.1605
3. Vazirani, Vijay (2001). Approximation algorithms. Berlin: Springer. pp. 74–83. ISBN   3540653678. OCLC   47097680.
4. Vazirani, Vijay V. (2003). Approximation Algorithms. Berlin: Springer. pp. 294–295. ISBN   3-540-65367-8.
5. H. Kellerer and U. Pferschy and D. Pisinger (2004). Knapsack Problems. Springer.CS1 maint: Uses authors parameter (link)
6. Jansen, Thomas (1998), "Introduction to the Theory of Complexity and Approximation Algorithms", in Mayr, Ernst W.; Prömel, Hans Jürgen; Steger, Angelika, Lectures on Proof Verification and Approximation Algorithms, Springer, pp. 5–28, doi:10.1007/BFb0053011, ISBN   9783540642015 . See discussion following Definition 1.30 on p. 20.


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Enter the sales and the profit into the Calculator. The calculator will evaluate the Expenses from Profit First.

## Expenses from Profit First Formula

S - P = E

Variables:

• E is the Expenses from Profit First ($) • S ($) is the sales
• P (\$) is the profit

To calculate Expenses from Profit First, subtract the profit from the sales.

## How to Calculate Expenses from Profit First?

The following steps outline how to calculate the Expenses from Profit First.

1. First, determine the sales.
2. Next, determine the profit.
3. Next, gather the formula from above = S – P = E.
4. Finally, calculate the Expenses from Profit First.
5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem :

Use the following variables as an example problem to test your knowledge.

sales = 30

profit = 20


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# Divide fractions and whole numbers word problems

Solve and interpret division word problems that include whole numbers and fractions.

### Problem

Andy painted at his house. He used start fraction, 1, divided by, 4, end fraction of his paint to paint 5 walls.
What fraction of his paint did Andy use on each wall if he used the same amount of paint on each?
of his paint


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Product Releases

# Signal Generators up to 6.6 GHz

Mon, 06/05/2006 - 11:44am

## LISTED UNDER:

National Instruments announces seven signal generators for applications from DC to microwave. The RF and microwave signal generators extend PXI to 6.6 GHz for continuous-wave (CW) sourcing, common tunable local oscillator (LO) and stimulus-response applications. All seven generators use direct digital synthesis (DDS) for high-resolution frequency hopping or phase-continuous sweeping for characterization applications. With the NI PXI-5652 6.6 GHz RF and microwave signal generator, the NI PXI-5651 3.3 GHz RF signal generator and the NI PXI-5650 1.3 GHz RF signal generator, engineers can set up sophisticated stimulus-response or tracking-generator applications using the NI PXI-5660 RF vector signal analyzer and the company’s LabVIEW graphical development environment. They are appropriate for clocking non-standard sample rates, such as WCDMA signals at baseband or intermediate frequency (IF).

Texas Instruments


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# linearmodels.panel.results.RandomEffectsResults.wald_test¶

RandomEffectsResults.wald_test(restriction=None, value=None, *, formula=None)

Test linear equality constraints using a Wald test

Parameters:
restriction

q by nvar array containing linear weights to apply to parameters when forming the restrictions. It is not possible to use both restriction and formula.

value

q element array containing the restricted values.

formulaUnion[str, list[str]]

formulaic linear constraints. The simplest formats are one of:

• A single comma-separated string such as “x1=0, x2+x3=1”

• A list of strings where each element is a single constraint such as [“x1=0”, “x2+x3=1”]

• A single string without commas to test simple constraints such as “x1=x2=x3=0”

• A dictionary where each key is a parameter restriction and the corresponding value is the restriction value, e.g., {“x1”: 0, “x2+x3”: 1}.

It is not possible to use both restriction and formula.

Returns:
WaldTestStatistic

Test statistic for null that restrictions are valid.

Notes

Hypothesis test examines whether $$H_0:C\theta=v$$ where the matrix C is restriction and v is value. The test statistic has a $$\chi^2_q$$ distribution where q is the number of rows in C.

Examples

>>> from linearmodels.datasets import wage_panel
>>> import statsmodels.api as sm
>>> import numpy as np
>>> import pandas as pd
>>> year = pd.Categorical(data.year)
>>> data = data.set_index(["nr", "year"])
>>> data["year"] = year
>>> from linearmodels.panel import PanelOLS
>>> exog_vars = ["expersq", "union", "married", "year"]

>>> mod = PanelOLS(data.lwage, exog, entity_effects=True)
>>> fe_res = mod.fit()

Test the restriction that union and married have 0 coefficients

>>> restriction = np.zeros((2, 11))
>>> restriction[0, 2] = 1
>>> restriction[1, 3] = 1
>>> value = np.array([0, 0])
>>> wald_res = fe_res.wald_test(restriction, value)

The same test using formulas

>>> formula = "union = married = 0"
>>> wald_res = fe_res.wald_test(formula=formula)

Return type:

WaldTestStatistic


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FIN 571 Week 3 Individual Practice Quiz

# FIN 571 Week 3 Individual Practice Quiz

A
0 points

FIN 571 Week 3 Individual Practice Quiz

University of Phoenix

1. The operating cycle

2. You are provided the following working capital information for the Ridge Company:

Ridge Company Account$Inventory$12,890 Accounts receivable12,800 Accounts payable12,670

Net sales$124,589 Cost of goods sold99,630 Cash conversion cycle: What is the cash conversion cycle for Ridge Company? 1. Ticktock Clocks sells 10,000 alarm clocks each year. If the total cost of placing an order is$65 and it costs $85 per year to carry the alarm clock in inventory, use the EOQ formula to calculate the optimal order size. 2. The asset substitution problem occurs when 3. M&M Proposition 1: Dynamo Corp. produces annual cash flows of$150 and is expected to exist forever. The company is currently financed with 75 percent equity and 25 percent debt. Your analysis tells you that the appropriate discount rates are 10 percent for the cash flows, and 7 percent for the debt. You currently own 10 percent of the stock.

How much are your cash flows today?

1. M&M Proposition 2: Melba's Toast has a capital structure with 30% debt and 70% equity. Its pretax cost of debt is 6%, and its cost of equity is 10%. The firm's marginal corporate income tax rate is 35%. What is the appropriate WACC?

2. According to the text, the financial plan covers a period of

3. The financing plan of a firm will indicate

4. Payout and retention ratio: Tradewinds Corp. has revenues of $9,651,220, costs of$6,080,412, interest payment of $511,233, and a tax rate of 34 percent. It paid dividends of$1,384,125 to shareholders. Find the firm's dividend payout ratio and retention ratio.

FIN 571
ajotatxe

A
0 points

#### Oh Snap! This Answer is Locked

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