The object initializer spreads properties from
defaults
and
unsafeOptions
source objects. The order in which the source objects are specified is important: later source object properties overwrite earlier ones.
Filling an incomplete object with default property values is an efficient strategy to make your code safe and durable. No matter the situation, the object always contains the full set of properties: and
undefined
cannot be generated.
The function parameters implicitly default to
undefined
.
Normally a function that is defined with a specific number of parameters should be invoked with the same number of arguments. In such case the parameters get the values you expect:
The invocation
multiply(5, 3)
makes the parameters
a
and
b
receive the corresponding
5
and
3
values. The multiplication is calculated as expected:
5 * 3 = 15
.
What does happen when you omit an argument on invocation? The parameter inside the function becomes
undefined
.
Let's slightly modify the previous example by calling the function with just one argument:
function multiply(a, b) { }
is defined with two parameters
a
and
b
. The invocation
multiply(5)
is performed with a single argument: as result
a
parameter is
5
, but
b
parameter is
undefined
.
Tip 6: Use default parameter value
Sometimes a function does not require the full set of arguments on invocation. You can simply set defaults for parameters that don't have a value.
Recalling the previous example, let's make an improvement. If
b
parameter is
undefined
, it gets assigned with a default value of
2
:
The function is invoked with a single argument
multiply(5)
. Initially
a
parameter is
2
and
b
is
undefined
. The conditional statement verifies whether
b
is
undefined
. If it happens,
b = 2
assignment sets a default value.
While the provided way to assign default values works, I don't recommend comparing directly against
undefined
. It's verbose and looks like a hack.
A better approach is to use the ES2015
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feature. It's short, expressive and no direct comparisons with
undefined
.
Modifying the previous example with a default parameter for
b
indeed looks great:
b = 2
in the function signature makes sure that if
b
is
undefined
, the parameter is defaulted to
2
.
Probability textbooks tend to be too simple, ignoring many important concepts and succumbing to the pedagogical issues we have discussed, or focus on the myriad technical details of probability theory and hence quickly fall beyond the proficiency of many readers. My favorite treatment of the more formal details of probability theory, and its predecessor measure theory, is Folland (1999) who spends significant time discussing concepts between the technical details.
From an abstract perspective, probability is a positive, conserved quantity which we want to distribute across a space, X . We take the total amount of this conserved quantity to be 1 with arbitrary units, but the mathematical consequences are the same regardless of this scaling. From this perspective probability is simply any abstract conserved quantity – in particular it does not refer to anything inherently random or uncertain.
A defines a mathematically self-consistent allocation of this conserved quantity across X . Letting A be a sufficiently well-defined subset of X , we write P π [ A ] as the probability assigned to A by the probability distribution π . Importantly, we want this allocation to be self-consistent – the allocation to any collection of disjoint sets, A n ∩ A m = 0 , n ≠ m , should be the same as the allocation to the union of those sets, P π [ ∪ N n = 1 A n ] = N ∑ n = 1 P π [ A n ] . In other words, no matter how we decompose the space X , or any well-defined subsets of X , we conserve probability.
For a finite collection of sets this self-consistency property is known as and would be sufficient if there were only a finite number of well-defined subsets in X . If we want to distribute probability across spaces with an infinite number of subsets, such as the real numbers, however, then we need to go a bit further and require self-consistency over any countable collection of disjoint sets, P π [ ∪ ∞ n = 1 A n ] = ∞ ∑ n = 1 P π [ A n ] . In particular, this property allows us to cover complex neighborhoods, such as that enclosed by a smooth surface, with an infinite collection of sets and then calculate the probability allocated to that neighborhood.
In addition to self-consistency we have to ensure that we assign all of the total probability in our allocation. This requires that all of the probability is allocated to the full space, P π [ X ] = 1 .
These three conditions completely specify a valid probability distribution, although to be formal we have to be careful about what we mean by “well-defined” subsets of X . Somewhat unnervingly we cannot construct an object that self-consistently allocates probability to subset of X because of some very weird, pathological subsets. Fortunately the same properties that make these subsets pathological also prevent them from belonging to any σ -algebra, consequently we can construct our probability distribution relative to a given σ -algebra, X .
Formally, then, probability theory is defined by , which we can write as:
The more familiar rules of probability theory can all be derived from these axioms. For example the last self-consistency condition implies that P π [ A ] + P π [ A c ] = P π [ X ] = 1 or P π [ A ] = 1 − P π [ A c ] .
A probability distribution is then completely specified by the ( X , X , π ) which is often denoted more compactly as x ∼ π where x ∈ X denotes the space, π denotes the probability distribution, and a valid σ -algebra is assumed.
The allocation of probability across a space immediately defines a way to summarize how functions of the form f : X → R behave. , E π [ f ] , reduce a function to a single real number by averaging the function output at every point, f ( x ) , weighted by the probability assigned around that point. This weighting process emphasizes how the function behaves in neighborhoods of high probability while diminishing its behavior in neighborhoods of low probability.
How exactly, however, do we formally construct these expectation values? The only expectation values that we can immediately calculate in closed form are the expectations of an that vanishes outside of a given set, I A [ x ] = { 1 , x ∈ A 0 , x ∉ A . The expectation of an indicator function is simply the weight assigned to A , which is just the probability allocated to that set, E π [ I A ] ≡ P π [ A ] . We can then build up the expectation value of an arbitrary function with a careful approximation in terms of these indicator functions in a process known as . For more detail see the following optional section.
When our space is a subset of the real line, X ⊆ R , there is a natural of X into R , ι : X → R x ↦ x . For example this embedding associates the natural numbers, { 0 , 1 , 2 , … } , with the corresponding values in the real line, or the interval [ 0 , 1 ] with the corresponding interval in the full real line.
In this circumstance we define the of the probability distribution as m π = E π [ ι ] , which quantifies the location around which the probability distribution is focusing its allocation. Similarly we define the of the probability distribution as V π = E π [ ( ι − m π ) 2 ] , which quantifies the breadth of the allocation around the mean. We will also refer to the variance of an arbitrary function as V π [ f ] = E π [ ( f − E π [ f ] ) 2 ] .
While we can always define expectation values of a function f : X → R , a probability distribution will not have a well-defined mean and variance unless there is some function whose expectation has a particular meaning. For example, if our space is a subset of the real numbers, X ⊆ R N , then there is no natural function whose expectation value defines a scalar mean. We can, however, define means and variances as expectations of the , ˆ x n : R N → R , that project a point x ∈ X onto each of the component axes. These component means and variances then provide some quantification of how the probability is allocated along each axis.
As we saw in
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only the indicator functions have immediate expectation values in terms of probabilities. In order to define expectation values of more general functions we have to build increasingly more complex functions out of these elementary ingredients.
The countable sum of indicator functions weighted by real numbers defines a , ϕ = ∑ n a n I A n . If we require that expectation is linear over this summation then the expectation value of any simple function is given by E π [ ϕ ] = E π [ ∑ n a n I A n ] = ∑ n a n E π [ I A n ] = ∑ n a n P π [ A n ] . Because of the countable additivity of π and the boundedness of probability, the expectation of a simple function will always be finite provided that each of the coefficients a n are themselves finite.
We can then use simple functions to approximate an everywhere-positive function, g : X → R + . A simple function with only a few terms defined over only a few sets will yield a poor approximation to g , but as we consider more terms and more sets we can build an increasingly accurate approximation. In particular, because of countable additivity we can construct a simple function bounded below by f that approximates f with arbitrary accuracy.
Consequently we define the expectation of an everywhere-positive function as the expectation of this approximating simple function. Because we were careful to consider only simple functions bounded by f we can also define the expectation of f as the largest expectation of all bounded simple functions, E π [ f ] = max
For functions that aren’t everywhere-positive we can decompose X into a collection of neighborhoods where f is entirely positive, A^{+}_{n} , and entirely negative, A^{-}_{m} . In those neighborhoods where f is entirely positive we apply the above procedure to define \mathbb{E}_{\pi} [ f \cdot \mathbb{I}_{A^{+}_{n}}], while in the neighborhoods where f is entirely negative we apply the above procedure on the negation of f to define \mathbb{E}_{\pi} [ -f \cdot \mathbb{I}_{A^{+}_{n}}]. . Those regions where g vanishes yield zero expectation values and can be ignored. We then define the expectation value of an arbitrary function g as the sum of these contributions, \mathbb{E}_{\pi} [ f ] = \sum_{n = 0}^{\infty} \mathbb{E}_{\pi} [ f \cdot \mathbb{I}_{A^{+}_{n}}] - \sum_{m = 0}^{\infty} \mathbb{E}_{\pi} [ -f \cdot \mathbb{I}_{A^{-}_{m}}].
Formally this procedure is known as and is a critical tool in the more general of which probability theory is a special case.
Once we have defined a probability distribution on a space, X , and a well-behaved collection of subsets, \mathcal{X} , we can then consider how the probability distribution transforms when X transforms. In particular, let f: X \rightarrow Y be a transformation from X to another space Y . Can this transformation also transform our probability distribution on X onto a probability distribution on Y , and if so under what conditions?
The answer is straightforward once we have selected a \sigma -algebra for Y as well, which we will denote \mathcal{Y} . In order for f to induce a probability distribution on Y we need the two \sigma -algebras to be compatible in some sense. In particular we need every subset B \in \mathcal{Y} to correspond to a unique subset f^{-1}(B) \in \mathcal{X} . If this holds for all subsets in \mathcal{Y} then we say that the transformation f is and we can define a distribution, \pi_{*} by \mathbb{P}_{\pi_{*}} [ B ] = \mathbb{P}_{\pi} [ f^{-1} (B) ]. In other words, if f is measurable then a self-consistent allocation of probability over X induces a self-consistent allocation of probability over Y .
One especially important class of measurable functions are those for which f(A) \in \mathcal{Y} for any A \in \mathcal{X} in addition to f^{-1}(B) \in \mathcal{X} for any B \in \mathcal{Y} . In this case f transforms not only a probability distribution on X into a probability distribution on Y but also a probability distribution on Y into a probability distribution on X . In this case we actually have one unique probability distribution that is just being defined over two different manifestations of the same abstract system. The two manifestations, for example, might correspond to different choices of coordinate system, or different choices of units, or different choices of language capable of the same descriptions. These transformations then serve as translations from one equivalent manifestation to another.
Measurable transformations can also be used to project a probability distribution over a space onto a probability distribution over a lower-dimensional subspace. Let \varpi: X \rightarrow Y be a that maps points in a space X to points in the subspace Y \subset X . It turns out that in this case a \sigma -algebra on X naturally defines a \sigma -algebra on Y and the projection operator is measurable with respect to this choice. Consequently any on X will transform into a unique on Y . More commonly we say that we the complementary subspace, Y^{C} .
Marginalization is a bit more straightforward when we are dealing with a product space, X \times Y , which is naturally equipped with the component projection operators \varpi_{X} : X \times Y \rightarrow X and \varpi_{Y}: X \times Y \rightarrow Y . In this case by pushing a distribution over (X \times Y, \mathcal{X} \times \mathcal{Y}) forwards along \varpi_{X} we marginalize out Y to give a probability distribution over (X, \mathcal{X}) . At the same time by pushing that same distribution forwards along \varpi_{Y} we can marginalize out X to give a probability distribution over (Y, \mathcal{Y}) .
Consider, for example, the three-dimensional space, \mathbb{R}^{3} , where the coordinate functions serve as projection operators onto the three axes, X , Y , and Z . Marginalizing out X transforms a probability distribution over X \times Y \times Z to give a probability distribution over the two-dimensional space, Y \times Z = \mathbb{R}^{2} . Marginalizing out Y then gives a probability distribution over the one-dimensional space, Z = \mathbb{R} .
As we saw in Section 2.4 , projection operators allow us to transform a probability distribution over a space to a probability distribution on some lower-dimensional subspace. Is it possible, however, to go the other way? Can we take a given marginal probability distribution on a subspace and construct a joint probability distribution on the total space that projects back to the marginal? We can if we can define an appropriate probability distribution on the complement of the given subspace.
Consider a N -dimensional , X , with the projection, \varpi : X \rightarrow Y , onto a K < N -dimensional , Y . By pushing a probability distribution on X along the projection operator we compress all of the information about how probability is distributed along the \varpi^{-1} (y) for each y \in Y . In order to reconstruct the original probability distribution from a marginal probability distribution we need to specify this lost information.
Every fiber takes the form of a N - K -dimensional space, F , and, like subspaces, these fiber spaces inherent a natural \sigma -algebra, \mathcal{F} , from the \sigma -algebra over the total space, \mathcal{X} . A defines a probability distribution over each fiber that varies with the base point, y , \begin{alignat*}{6} \mathbb{P}_{F \mid Y} :\; \mathcal{F} \times Y \rightarrow \; [0, 1] \\ (A, y) \mapsto \mathbb{P}_{F \mid Y} [A, y]. \end{alignat*} Evaluated at any y \in Y the conditional probability distribution defines a probability distribution over the corresponding fiber space, (F, \mathcal{F}) . On the other hand, when evaluated at a given subset A \in \mathcal{F} the conditional probability distribution becomes a measurable function from Y into [0, 1] that quantifies how the probability of that set varies as we move from one fiber to the next.
Given a marginal distribution, \pi_{Y} , we can then define a probability distribution over the total space by taking an expectation value, \mathbb{P}_{X} [ A ] = \mathbb{E}_{Y} [ \mathbb{P}_{F \mid Y} [A \cap \varpi^{-1} (y), y] ].
The induced joint distribution on the total space is consistent in the sense that if we transform it back along the projection operator we recover the marginal distribution with which we started.
This construction becomes significantly easier when we consider a product space, X \times Y and the projection \varpi: X \times Y \rightarrow Y . In this case the fiber space is just X .
The conditional probability distribution becomes \begin{alignat*}{6} \mathbb{P}_{X \mid Y} :\; \mathcal{X} \times Y \rightarrow \; [0, 1] \\ (A, y) \mapsto \mathbb{P}_{X \mid Y}[A, y]. \end{alignat*} with joint distribution \mathbb{P}_{X \times Y} [ A ] = \mathbb{E}_{Y} [ \mathbb{P}_{X \mid Y} [A \cap X, y] ].
Conditional probability distributions are especially useful when we want to construct a complex probability distribution over a high-dimensional space. We can reduce the specification of the ungainly joint probability distribution with a sequence of lower-dimensional conditional probability distributions and marginal probability distributions about which we can more easily reason. In the context of modeling an observational process, this method of construction a complicated distribution from intermediate conditional probability distributions is known as modeling. In particular, each intermediate conditional probability distribution models some fragment of the full observational process.
As we saw in the previous section, formal probability theory is simply the study of probability distributions that allocate a finite, conserved quantity across a space, the expectation values that such an allocation induces, and how the allocation behaves under transformations of the underlying space. While there is myriad complexity in the details of that study, the basics concepts are relatively straightforward.
This page helps you design your Apache Beam pipeline. It includes information about how to determine your pipeline’s structure, how to choose which transforms to apply to your data, and how to determine your input and output methods.
Before reading this section, it is recommended that you become familiar with the information in the
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.
When designing your Beam pipeline, consider a few basic questions:
Where is your input data stored? What does your data look like? What do you want to do with your data? What does your output data look like, and where should it go?The simplest pipelines represent a linear flow of operations, as shown in figure 1.
Figure 1: A linear pipeline.
However, your pipeline can be significantly more complex. A pipeline represents a
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of steps. It can have multiple input sources, multiple output sinks, and its operations (
PTransform
s) can both read and output multiple
PCollection
s. The following examples show some of the different shapes your pipeline can take.
It’s important to understand that transforms do not consume
PCollection
s; instead, they consider each individual element of a
PCollection
and create a new
PCollection
as output. This way, you can do different things to different elements in the same
PCollection
.
You can use the same
PCollection
as input for multiple transforms without consuming the input or altering it.
The pipeline in figure 2 is a branching pipeline. The pipeline reads its input (first names represented as strings) from a database table and creates a
PCollection
of table rows. Then, the pipeline applies multiple transforms to the
same
PCollection
. Transform A extracts all the names in that
PCollection
that start with the letter ‘A’, and Transform B extracts all the names in that
PCollection
that start with the letter ‘B’. Both transforms A and B have the same input
PCollection
.
Figure 2: A branching pipeline. Two transforms are applied to a single PCollection of database table rows.
The following example code applies two transforms to a single input collection.
Another way to branch a pipeline is to have a
single
transform output to multiple
PCollection
s by using
tagged outputs
. Transforms that produce more than one output process each element of the input once, and output to zero or more
PCollection
s.
Figure 3 illustrates the same example described above, but with one transform that produces multiple outputs. Names that start with ‘A’ are added to the main output
PCollection
, and names that start with ‘B’ are added to an additional output
PCollection
.
“ The Maury Island Incident ” – a short film that was shot locally in the Burien area last summer – has been turned into a 6-part original series and will premiere on Seattle-based IndieFlix , an independent film streaming service on Aug. 19, 2014.
Based on declassified FBI documents, the film tells the incredible, tragic, and forgotten story of Harold Dahl, who on June 21, 1947, alleged a UFO sighting over Puget Sound, Washington. This sparked ‘the summer of the saucers,’ the modern era of UFO obsession, the first appearance of a ‘Man in Black’ as well as a governmental battle over UFO sighting jurisdiction reaching directly to FBI Executive Director J. Edgar Hoover.
The Aug. 19 date coincides with an FBI document sent to Executive Director J. Edgar Hoover, explaining how Dahl’s original claim that the sighting was a hoax was only said to avoid any further damage to his family. This historic document will also be released on Aug. 19 by filmmakers as a downloadable PDF on the official website www.mauryislandincident.com .
www.mauryislandincident.com“IndieFlix viewers will not only learn new information about a lost, historic UFO case, they’ll also find out just how interested J. Edgar Hoover was with these ‘flying disc’ occurrences,” Producer/Director Scott Schaefer said. “And we will also be releasing some fascinating declassified FBI documents that show his personal interest in UFOs, specifically The Maury Island Incident.”
Initially shot as a short in the south Puget Sound area, with local talent and crew, The Maury Island Incident has been a labor of love for Producer/Writer Edmiston and Producer/Director Schaefer, and Washington FilmWorks, which gave out an Innovation Lab Award to this production. The Lab is a groundbreaking new program offering funding assistance to Washington filmmakers and filmmakers using emerging technologies. In its comments, the jury said this of the film:
“Equal parts mystery and documentary, The Maury Island Incident exposes a fascinating hidden history: the first recorded UFO incidents in the US didn’t occur in Roswell, but in Washington. This captivating project brings a spooky Seattle area legend to light and sets the stage for ongoing storytelling on the subject.”
Thanks to the help and support of Washington Filmworks’ Innovation Lab, Edmiston and Schaefer got the opportunity to turn their content into a series.
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