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From Alan Woodward <romseyg...@gmail.com>
Subject Re: lucene-solr:master: Squashed commit of the following:
Date Wed, 09 May 2018 20:08:20 GMT
I think this has broken precommit?


build-nav-data-files:
     [java] Building up tree of all known pages
     [java] ERROR: Orphan page: /Users/romseygeek/projects/lucene-solr/solr/build/solr-ref-guide/content/dsp.adoc
     [java] Exception in thread "main" java.lang.RuntimeException: Found 1 orphan pages (which
are not in the 'page-children' attribute of any other pages)
     [java] 	at BuildNavAndPDFBody.main(BuildNavAndPDFBody.java:82)

> On 9 May 2018, at 18:24, jbernste@apache.org wrote:
> 
> Repository: lucene-solr
> Updated Branches:
>  refs/heads/master 4177252a1 -> 144f00a1e
> 
> 
> Squashed commit of the following:
> 
> commit e5074c3223e394af17f686294a67a1dd3ecdd147
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Wed May 9 13:16:34 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 69cdeccf161177d10f4d2407542392aaee3fcfe8
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Wed May 9 13:08:02 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit c94f0c87c3e57c023d622ad2411e522c4aac491c
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Wed May 9 11:54:58 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 68dd1e73355cb84410f2d0ff3a51797ed6194a10
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Wed May 9 10:54:32 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 04a010543418a469100fa299c606a7b1eed452e1
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Wed May 9 10:47:27 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit a6bbfbadaafe33fcdf93d5c72755e30dadadf017
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Wed May 9 09:40:08 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 5d27961aa291bcd71527337632981bcdf62369b4
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 20:43:33 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit e982cf939f429c05b736f6292c68dd96d7ebc027
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 13:27:29 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit aae78ab6f387c28a080021bc919ef51864540be2
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 12:23:52 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 0787ad76f0f4c62c860784b15490d8a988939997
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 12:20:38 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 4df098376ba05188702cca8582959c3fe18066f5
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 12:12:11 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 5c0be5136bbab7e0c33b3b8a7b0395b1b330e96d
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 12:04:57 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 6c6feac4c2e5a49a5eab87a228713d1b93c8fc70
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 11:57:49 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 7d46d11c9dd3a51b68600c2c889f586147545294
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Tue May 8 11:50:51 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 8b6bf19d0091203ed63b39d070dd02a9bece6a61
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Mon May 7 10:53:14 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit 5466591999816eaacde6ce18d824d7688e5f4fe8
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Fri May 4 15:12:43 2018 -0400
> 
>    SOLR-12280: WIP
> 
> commit d7fff7d557a7fd26011c21445b7969b2cd81036f
> Author: Joel Bernstein <jbernste@apache.org>
> Date:   Fri Apr 27 12:50:27 2018 -0400
> 
>    SOLR-12280: Initial commit
> 
> 
> Project: http://git-wip-us.apache.org/repos/asf/lucene-solr/repo
> Commit: http://git-wip-us.apache.org/repos/asf/lucene-solr/commit/144f00a1
> Tree: http://git-wip-us.apache.org/repos/asf/lucene-solr/tree/144f00a1
> Diff: http://git-wip-us.apache.org/repos/asf/lucene-solr/diff/144f00a1
> 
> Branch: refs/heads/master
> Commit: 144f00a1e315541a28f526f4cdf1e55eb60c862b
> Parents: 4177252
> Author: Joel Bernstein <jbernste@apache.org>
> Authored: Wed May 9 13:24:08 2018 -0400
> Committer: Joel Bernstein <jbernste@apache.org>
> Committed: Wed May 9 13:24:08 2018 -0400
> 
> ----------------------------------------------------------------------
> solr/solr-ref-guide/src/dsp.adoc                | 719 +++++++++++++++++++
> .../hidden-signal-autocorrelation.png           | Bin 0 -> 258831 bytes
> .../math-expressions/hidden-signal-fft.png      | Bin 0 -> 215981 bytes
> .../images/math-expressions/hidden-signal.png   | Bin 0 -> 319100 bytes
> .../math-expressions/noise-autocorrelation.png  | Bin 0 -> 204511 bytes
> .../src/images/math-expressions/noise-fft.png   | Bin 0 -> 319551 bytes
> .../src/images/math-expressions/noise.png       | Bin 0 -> 375565 bytes
> .../math-expressions/signal-autocorrelation.png | Bin 0 -> 322164 bytes
> .../src/images/math-expressions/signal-fft.png  | Bin 0 -> 140111 bytes
> .../src/images/math-expressions/signal.png      | Bin 0 -> 365018 bytes
> solr/solr-ref-guide/src/math-expressions.adoc   |   2 +
> 11 files changed, 721 insertions(+)
> ----------------------------------------------------------------------
> 
> 
> http://git-wip-us.apache.org/repos/asf/lucene-solr/blob/144f00a1/solr/solr-ref-guide/src/dsp.adoc
> ----------------------------------------------------------------------
> diff --git a/solr/solr-ref-guide/src/dsp.adoc b/solr/solr-ref-guide/src/dsp.adoc
> new file mode 100644
> index 0000000..348030e
> --- /dev/null
> +++ b/solr/solr-ref-guide/src/dsp.adoc
> @@ -0,0 +1,719 @@
> += Digital Signal Processing
> +// Licensed to the Apache Software Foundation (ASF) under one
> +// or more contributor license agreements.  See the NOTICE file
> +// distributed with this work for additional information
> +// regarding copyright ownership.  The ASF licenses this file
> +// to you under the Apache License, Version 2.0 (the
> +// "License"); you may not use this file except in compliance
> +// with the License.  You may obtain a copy of the License at
> +//
> +//   http://www.apache.org/licenses/LICENSE-2.0
> +//
> +// Unless required by applicable law or agreed to in writing,
> +// software distributed under the License is distributed on an
> +// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
> +// KIND, either express or implied.  See the License for the
> +// specific language governing permissions and limitations
> +// under the License.
> +
> +This section of the user guide explores functions that are commonly used in the field
of
> +Digital Signal Processing (DSP).
> +
> +== Dot product
> +
> +The `dotProduct` function is used to calculate the dot product of two arrays.
> +The dot product is a fundamental calculation for the DSP functions discussed in this
section. Before diving into
> +the more advanced DSP functions, its useful to get a better understanding of how the
dot product calculation works.
> +
> +=== Combining two arrays
> +
> +The `dotProduct` function can be used to combine two arrays into a single product. A
simple example can help
> +illustrate this concept.
> +
> +In the example below two arrays are set to variables *a* and *b* and then operated on
by the `dotProduct` function.
> +The output of the `dotProduct` function is set to variable *c*.
> +
> +Then the `mean` function is then used to compute the mean of the first array which is
set to the variable `d`.
> +
> +Both the *dot product* and the *mean* are included in the output.
> +
> +When we look at the output of this expression we see that the *dot product* and the
*mean* of the first array
> +are both 30.
> +
> +The dot product function *calculated the mean* of the first array.
> +
> +[source,text]
> +----
> +let(echo="c, d",
> +    a=array(10, 20, 30, 40, 50),
> +    b=array(.2, .2, .2, .2, .2),
> +    c=dotProduct(a, b),
> +    d=mean(a))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": 30,
> +        "d": 30
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +To get a better understanding of how the dot product calculated the mean we can perform
the steps of the
> +calculation using vector math and look at the output of each step.
> +
> +In the example below the `ebeMultiply` function performs an element-by-element multiplication
of
> +two arrays. This is the first step of the dot product calculation. The result of the
element-by-element
> +multiplication is assigned to variable *c*.
> +
> +In the next step the `add` function adds all the elements of the array in variable *c*.
> +
> +Notice that multiplying each element of the first array by .2 and then adding the results
is
> +equivalent to the formula for computing the mean of the first array. The formula for
computing the mean
> +of an array is to add all the elements and divide by the number of elements.
> +
> +The output includes the output of both the `ebeMultiply` function and the `add` function.
> +
> +[source,text]
> +----
> +let(echo="c, d",
> +    a=array(10, 20, 30, 40, 50),
> +    b=array(.2, .2, .2, .2, .2),
> +    c=ebeMultiply(a, b),
> +    d=add(c))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": [
> +          2,
> +          4,
> +          6,
> +          8,
> +          10
> +        ],
> +        "d": 30
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +In the example above two arrays were combined in a way that produced the mean of the
first. In the second array
> +each value was set to .2. Another way of looking at this is that each value in the second
array has the same weight.
> +By varying the weights in the second array we can produce a different result. For example
if the first array represents a time series,
> +the weights in the second array can be set to add more weight to a particular element
in the first array.
> +
> +The example below creates a weighted average with the weight decreasing from right to
left. Notice that the weighted mean
> +of 36.666 is larger than the previous mean which was 30. This is because more weight
was given to last element in the
> +array.
> +
> +[source,text]
> +----
> +let(echo="c, d",
> +    a=array(10, 20, 30, 40, 50),
> +    b=array(.066666666666666,.133333333333333,.2, .266666666666666, .33333333333333),
> +    c=ebeMultiply(a, b),
> +    d=add(c))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": [
> +          0.66666666666666,
> +          2.66666666666666,
> +          6,
> +          10.66666666666664,
> +          16.6666666666665
> +        ],
> +        "d": 36.66666666666646
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +=== Representing Correlation
> +
> +Often when we think of correlation, we are thinking of *Pearsons* correlation in the
field of statistics. But the definition of
> +correlation is actually more general: a mutual relationship or connection between two
or more things.
> +In the field of digital signal processing the dot product is used to represent correlation.
The examples below demonstrates
> +how the dot product can be used to represent correlation.
> +
> +In the example below the dot product is computed for two vectors. Notice that the vectors
have different values that fluctuate
> +together. The output of the dot product is 190, which is hard to reason about because
because its not scaled.
> +
> +[source,text]
> +----
> +let(echo="c, d",
> +    a=array(10, 20, 30, 20, 10),
> +    b=array(1, 2, 3, 2, 1),
> +    c=dotProduct(a, b))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": 190
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +One approach to scaling the dot product is to first scale the vectors so that both vectors
have a magnitude of 1. Vectors with a
> +magnitude of 1, also called unit vectors, are used when comparing only the angle between
vectors rather then the magnitude.
> +The `unitize` function can be used to unitize the vectors before calculating the dot
product.
> +
> +Notice in the example below the dot product result, set to variable *e*, is effectively
1. When applied to unit vectors the dot product
> +will be scaled between 1 and -1. Also notice in the example `cosineSimilarity` is calculated
on the *unscaled* vectors and the
> +answer is also effectively 1. This is because *cosine similarity* is a scaled *dot product*.
> +
> +
> +[source,text]
> +----
> +let(echo="e, f",
> +    a=array(10, 20, 30, 20, 10),
> +    b=array(1, 2, 3, 2, 1),
> +    c=unitize(a),
> +    d=unitize(b),
> +    e=dotProduct(c, d),
> +    f=cosineSimilarity(a, b))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "e": 0.9999999999999998,
> +        "f": 0.9999999999999999
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +If we transpose the first two numbers in the first array, so that the vectors
> +are not perfectly correlated, we see that the cosine similarity drops. This illustrates
> +how the dot product represents correlation.
> +
> +[source,text]
> +----
> +let(echo="c, d",
> +    a=array(20, 10, 30, 20, 10),
> +    b=array(1, 2, 3, 2, 1),
> +    c=cosineSimilarity(a, b))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": 0.9473684210526314
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +== Convolution
> +
> +The `conv` function calculates the convolution of two vectors. The convolution is calculated
by *reversing*
> +the second vector and sliding it across the first vector. The *dot product* of the two
vectors
> +is calculated at each point as the second vector is slid across the first vector.
> +The dot products are collected in a *third vector* which is the *convolution* of the
two vectors.
> +
> +=== Moving Average
> +
> +Before looking at an example of convolution its useful to review the `movingAvg` function.
The moving average
> +function computes a moving average by sliding a window across a vector and computing
> +the average of the window at each shift. If that sounds similar to convolution, that's
because the `movingAvg` function
> +is syntactic sugar for convolution.
> +
> +Below is an example of a moving average with a window size of 5. Notice that original
vector has 13 elements
> +but the result of the moving average has only 9 elements. This is because the `movingAvg`
function
> +only begins generating results when it has a full window. In this case because the window
size is 5 so the
> +moving average starts generating results from the 4th index of the original array.
> +
> +[source,text]
> +----
> +let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    b=movingAvg(a, 5))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "b": [
> +          3,
> +          4,
> +          5,
> +          5.6,
> +          5.8,
> +          5.6,
> +          5,
> +          4,
> +          3
> +        ]
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +=== Convolutional Smoothing
> +
> +The moving average can also be computed using convolution. In the example
> +below the `conv` function is used to compute the moving average of the first array
> +by applying the second array as the filter.
> +
> +Looking at the result, we see that it is not exactly the same as the result
> +of the `movingAvg` function. That is because the `conv` pads zeros
> +to the front and back of the first vector so that the window size is always full.
> +
> +[source,text]
> +----
> +let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    b=array(.2, .2, .2, .2, .2),
> +    c=conv(a, b))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": [
> +          0.2,
> +          0.6000000000000001,
> +          1.2,
> +          2.0000000000000004,
> +          3.0000000000000004,
> +          4,
> +          5,
> +          5.6000000000000005,
> +          5.800000000000001,
> +          5.6000000000000005,
> +          5.000000000000001,
> +          4,
> +          3,
> +          2,
> +          1.2000000000000002,
> +          0.6000000000000001,
> +          0.2
> +        ]
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +We achieve the same result as the `movingAvg` gunction by using the `copyOfRange` function
to copy a range of
> +the result that drops the first and last 4 values of
> +the convolution result. In the example below the `precision` function is also also used
to remove floating point errors from the
> +convolution result. When this is added the output is exactly the same as the `movingAvg`
function.
> +
> +[source,text]
> +----
> +let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    b=array(.2, .2, .2, .2, .2),
> +    c=conv(a, b),
> +    d=copyOfRange(c, 4, 13),
> +    e=precision(d, 2))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "e": [
> +          3,
> +          4,
> +          5,
> +          5.6,
> +          5.8,
> +          5.6,
> +          5,
> +          4,
> +          3
> +        ]
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +== Cross-Correlation
> +
> +Cross-correlation is used to determine the delay between two signals. This is accomplished
by sliding one signal across another
> +and calculating the dot product at each shift. The dot products are collected into a
vector which represents the correlation
> +at each shift. The highest dot product in the cross-correlation vector is the point
where the two signals are most closely correlated.
> +
> +The sliding dot product used in convolution can also be used to represent cross-correlation
between two vectors. The only
> +difference in the formula when representing correlation is that the second vector is
*not reversed*.
> +
> +Notice in the example below that the second vector is reversed by the `rev` function
before it is operated on by the `conv` function.
> +The `conv` function reverses the second vector so it will be flipped back to its original
order to perform the correlation calculation
> +rather then the convolution calculation.
> +
> +Notice in the result the highest value is 217. This is the point where the two vectors
have the highest correlation.
> +
> +[source,text]
> +----
> +let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    b=array(4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    c=conv(a, rev(b)))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": [
> +          1,
> +          4,
> +          10,
> +          20,
> +          35,
> +          56,
> +          84,
> +          116,
> +          149,
> +          180,
> +          203,
> +          216,
> +          217,
> +          204,
> +          180,
> +          148,
> +          111,
> +          78,
> +          50,
> +          28,
> +          13,
> +          4
> +        ]
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +== Find Delay
> +
> +It is fairly simple to compute the delay from the cross-correlation result, but a convenience
function called `finddelay` can
> +be used to find the delay directly. Under the covers `finddelay` uses convolutional
math to compute the cross-correlation vector
> +and then computes the delay between the two signals.
> +
> +Below is an example of the `finddelay` function. Notice that the `finddelay` function
reports a 3 period delay between the first
> +and second signal.
> +
> +[source,text]
> +----
> +let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    b=array(4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
> +    c=finddelay(a, b))
> +----
> +
> +When this expression is sent to the /stream handler it responds with:
> +
> +[source,json]
> +----
> +{
> +  "result-set": {
> +    "docs": [
> +      {
> +        "c": 3
> +      },
> +      {
> +        "EOF": true,
> +        "RESPONSE_TIME": 0
> +      }
> +    ]
> +  }
> +}
> +----
> +
> +== Autocorrelation
> +
> +Autocorrelation measures the degree to which a signal is correlated with itself. Autocorrelation
is used to determine
> +if a vector contains a signal or is purely random.
> +
> +A few examples, with plots, will help to understand the concepts.
> +
> +In the first example the `sin` function is wrapped around a `sequence` function to generate
a sine wave. The result of this
> +is plotted in the image below. Notice that there is a structure to the plot that is
clearly not random.
> +
> +[source,text]
> +----
> +sin(sequence(256, 0, 6))
> +----
> +
> +image::images/math-expressions/signal.png[]
> +
> +
> +In the next example the `sample` function is used to draw 256 samples from a `uniformDistribution`
to create a
> +vector of random data. The result of this is plotted in the image below. Notice that
there is no clear structure to the
> +data and the data appears to be random.
> +
> +[source,text]
> +----
> +sample(uniformDistribution(-1.5, 1.5), 256)
> +----
> +
> +image::images/math-expressions/noise.png[]
> +
> +
> +In the next example the random noise is added to the sine wave using the `ebeAdd` function.
> +The result of this is plotted in the image below. Notice that the sine wave has been
hidden
> +somewhat within the noise. Its difficult to say for sure if there is structure. As plots
> +becomes more dense it can become harder to see a pattern hidden within noise.
> +
> +[source,text]
> +----
> +let(a=sin(sequence(256, 0, 6)),
> +    b=sample(uniformDistribution(-1.5, 1.5), 256),
> +    c=ebeAdd(a, b))
> +----
> +
> +image::images/math-expressions/hidden-signal.png[]
> +
> +
> +In the next examples autocorrelation is performed with each of the vectors shown above
to see what the
> +autocorrelation plots look like.
> +
> +In the example below the `conv` function is used to autocorrelate the first vector which
is the sine wave.
> +Notice that the `conv` function is simply correlating the sine wave with itself.
> +
> +The plot has a very distinct structure to it. As the sine wave is slid across a copy
of itself the correlation
> +moves up and down in increasing intensity until it reaches a peak. This peak is directly
in the center and is the
> +the point where the sine waves are directly lined up. Following the peak the correlation
moves up and down in decreasing
> +intensity as the sine wave slides farther away from being directly lined up.
> +
> +This is the autocorrelation plot of a pure signal.
> +
> +[source,text]
> +----
> +let(a=sin(sequence(256, 0, 6)),
> +    b=conv(a, rev(a)),
> +----
> +
> +image::images/math-expressions/signal-autocorrelation.png[]
> +
> +
> +In the example below autocorrelation is performed with the vector of pure noise. Notice
that the autocorrelation
> +plot has a very different plot then the sine wave. In this plot there is long period
of low intensity correlation that appears
> +to be random. Then in the center a peak of high intensity correlation where the vectors
are directly lined up.
> +This is followed by another long period of low intensity correlation.
> +
> +This is the autocorrelation plot of pure noise.
> +
> +[source,text]
> +----
> +let(a=sample(uniformDistribution(-1.5, 1.5), 256),
> +    b=conv(a, rev(a)),
> +----
> +
> +image::images/math-expressions/noise-autocorrelation.png[]
> +
> +
> +In the example below autocorrelation is performed on the vector with the sine wave hidden
within the noise.
> +Notice that this plot shows very clear signs of structure which is similar to autocorrelation
plot of the
> +pure signal. The correlation is less intense due to noise but the shape of the correlation
plot suggests
> +strongly that there is an underlying signal hidden within the noise.
> +
> +[source,text]
> +----
> +let(a=sin(sequence(256, 0, 6)),
> +    b=sample(uniformDistribution(-1.5, 1.5), 256),
> +    c=ebeAdd(a, b),
> +    d=conv(c, rev(c))
> +----
> +
> +image::images/math-expressions/hidden-signal-autocorrelation.png[]
> +
> +
> +== Discrete Fourier Transform
> +
> +The convolution based functions described above are operating on signals in the time
domain. In the time
> +domain the X axis is time and the Y axis is the quantity of some value at a specific
point in time.
> +
> +The discrete Fourier Transform translates a time domain signal into the frequency domain.
> +In the frequency domain the X axis is frequency, and Y axis is the accumulated power
at a specific frequency.
> +
> +The basic principle is that every time domain signal is composed of one or more signals
(sine waves)
> +at different frequencies. The discrete Fourier transform decomposes a time domain signal
into its component
> +frequencies and measures the power at each frequency.
> +
> +The discrete Fourier transform has many important uses. In the example below, the discrete
Fourier transform is used
> +to determine if a signal has structure or if it is purely random.
> +
> +=== Complex Result
> +
> +The `fft` function performs the discrete Fourier Transform on a vector of *real* data.
The result
> +of the `fft` function is returned as *complex* numbers. A complex number has two parts,
*real* and *imaginary*.
> +The imaginary part of the complex number is ignored in the examples below, but there
> +are many tutorials on the FFT and that include complex numbers available online.
> +
> +But before diving into the examples it is important to understand how the `fft` function
formats the
> +complex numbers in the result.
> +
> +The `fft` function returns a `matrix` with two rows. The first row in the matrix is
the *real*
> +part of the complex result. The second row in the matrix is the *imaginary* part of
the complex result.
> +
> +The `rowAt` function can be used to access the rows so they can be processed as vectors.
> +This approach was taken because all of the vector math functions operate on vectors
of real numbers.
> +Rather then introducing a complex number abstraction into the expression language, the
`fft` result is
> +represented as two vectors of real numbers.
> +
> +=== Fast Fourier Transform Examples
> +
> +In the first example the `fft` function is called on the sine wave used in the autocorrelation
example.
> +
> +The results of the `fft` function is a matrix. The `rowAt` function is used to return
the first row of
> +the matrix which is a vector containing the real values of the fft response.
> +
> +The plot of the real values of the `fft` response is shown below. Notice there are two
> +peaks on opposite sides of the plot. The plot is actually showing a mirrored response.
The right side
> +of the plot is an exact mirror of the left side. This is expected when the `fft` is
run on real rather then
> +complex data.
> +
> +Also notice that the `fft` has accumulated significant power in a single peak. This
is the power associated with
> +the specific frequency of the sine wave. The vast majority of frequencies in the plot
have close to 0 power
> +associated with them. This `fft` shows a clear signal with very low levels of noise.
> +
> +[source,text]
> +----
> +let(a=sin(sequence(256, 0, 6)),
> +    b=fft(a),
> +    c=rowAt(b, 0))
> +----
> +
> +
> +image::images/math-expressions/signal-fft.png[]
> +
> +In the second example the `fft` function is called on a vector of random data similar
to one used in the
> +autocorrelation example. The plot of the real values of the `fft` response is shown
below.
> +
> +Notice that in is this response there is no clear peak. Instead all frequencies have
accumulated a random level of
> +power. This `fft` shows no clear sign of signal and appears to be noise.
> +
> +
> +[source,text]
> +----
> +let(a=sample(uniformDistribution(-1.5, 1.5), 256),
> +    b=fft(a),
> +    c=rowAt(b, 0))
> +----
> +
> +image::images/math-expressions/noise-fft.png[]
> +
> +
> +In the third example the `fft` function is called on the same signal hidden within noise
that was used for
> +the autocorrelation example. The plot of the real values of the `fft` response is shown
below.
> +
> +Notice that there are two clear mirrored peaks, at the same locations as the `fft` of
the pure signal. But
> +there is also now considerable noise on the frequencies. The `fft` has found the signal
and but also
> +shows that there is considerable noise along with the signal.
> +
> +[source,text]
> +----
> +let(a=sin(sequence(256, 0, 6)),
> +    b=sample(uniformDistribution(-1.5, 1.5), 256),
> +    c=ebeAdd(a, b),
> +    d=fft(c),
> +    e=rowAt(d, 0))
> +----
> +
> +image::images/math-expressions/hidden-signal-fft.png[]
> +
> +
> 
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> http://git-wip-us.apache.org/repos/asf/lucene-solr/blob/144f00a1/solr/solr-ref-guide/src/math-expressions.adoc
> ----------------------------------------------------------------------
> diff --git a/solr/solr-ref-guide/src/math-expressions.adoc b/solr/solr-ref-guide/src/math-expressions.adoc
> index 3f9ed1a..0b8fafc 100644
> --- a/solr/solr-ref-guide/src/math-expressions.adoc
> +++ b/solr/solr-ref-guide/src/math-expressions.adoc
> @@ -56,4 +56,6 @@ record in your Solr Cloud cluster computable.
> 
> == <<curve-fitting.adoc#curve-fitting,Curve Fitting>>
> 
> +== <<dsp.adoc#digital-signal-processing, Digital Signal Processing>>
> +
> == <<machine-learning.adoc#machine-learning,Machine Learning>>
> 


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