Pandas dataframe, German vocabulary – select words by matching a few 3-char-grams – IV

In the last posts of this mini-series we have studied if and how we can use three 3-char-grams at defined positions of a string token to identify matching words in a reference vocabulary. We have seen that we should choose some distance between the char-grams and that we should use the words length information to keep the list of possible hits small.

Such a search may be interesting if there is only fragmented information available about some words of a text or if one cannot trust the whole token to be written correctly. There may be other applications. Note: This has so far nothing to do with text analysis based on machine learning procedures. I would put the whole topic more in the field of text preparation or text rebuilding. But, I think that one can combine our simple identification of fitting words by 3-char-grams with ML-methods which evaluate the similarity or distance of a (possibly misspelled) token with vocabulary words: When we get a long hit-list we could invoke ML-methods to to determine the best fitting word.

We saw that we can do a 100,000 search runs with 3-char-grams on a decent vocabulary of around 2 million words in a Pandas dataframe below a 1.3 minutes on one CPU core of an older PC. In this concluding article I want to look a bit at the idea of multiprocessing the search with up to 4 CPU cores.

Points to take into account when using multiprocessing - do not expect too much

Pandas normally just involves one CPU core to do its job. And not all operations on a Pandas dataframe may be well suited for multiprocessing. Readers who have followed the code fragments in this series so far will probably and rightly assume that there is indeed a chance for reasonably separating our search process for words or at least major parts of it.

But even then - there is always some overhead to expect from splitting a Pandas dataframe into segments (or "partitions") for a separate operations on different CPU cores. Overhead is also expected from the task to correctly to combine the particular results from the different processor cores to a data unity (here: dataframe) again at the end of a multiprocessed run.

A bottleneck for multiprocessing may also arise if multiple processes have to access certain distinct objects in memory at the same time. In our case we this point is to be expected for the access of and search within distinct sub-dataframes of the vocabulary containing words of a specific length.

Due to overhead and bottlenecks we do not expect that a certain problem scales directly and linearly with the number of CPU cores. Another point is that although the Linux OS may recognize a hyperthreading physical core of an Intel processor as two cores - but it may not be able to use such virtual cores in a given context as if they were real separate physical cores.

Code to invoke multiple processor cores

In this article I just use the standard Python "multiprocessing" module. (I did not test Ray yet - as a first trial gave me trouble in some preparing code-segments of my Jupyter notebooks. I did not have time to solve the problems there.)

Following some advice on the Internet I handled parallelization in the following way:

import multiprocessing
from multiprocessing import cpu_count, Pool

#cores = cpu_count() # Number of physical CPU cores on your system
cores = 4
partitions = cores # But actually you can define as many partitions as you want

def parallelize(data, func):
    data_split = np.array_split(data, partitions)
    pool = Pool(cores)
    data = pd.concat(pool.map(func, data_split), copy=False)
    pool.close()
    pool.join()
    return data

The basic function, corresponding to the parameter "func" of function "parallelize", which shall be executed in our case is structurally well known from the last posts of this article series:

We perform a search via putting conditions on columns (of the vocabulary-dataframe) containing 3-char-grams at different positions. The search is done on sub-dataframes of the vocabulary containing only words with a given length. The respective addresses are controlled by a Python dictionary "d_df"; see the last post for its creation. We then build a list of indices of fitting words. The dataframe containing the test tokens - in our case a random selection of real vocabulary words - will be called "dfw" inside the function "func() => getlen()" (see below). To understand the code you should be aware of the fact that the original dataframe is split into (4) partitions.

We only return the length of the list of hits and not the list of indices for each token itself.

# Function for parallelized operation 
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
def getlen(dfw):
    # Note 1: The dfw passed is a segment ("partition") of the original dataframe  
    # Note 2: We use a dict d_lilen which was defined outside  
    #         and is under the control of the parallelization manager
    
    num_rows = len(dfw)
    for i in range(0, num_rows):
        len_w = dfw.iat[i,0]
        idx = dfw.iat[i,33]
        
        df_name = "df_" + str(len_w)
        df_ = d_df[df_name]

        j_m = math.floor(len_w/2)+1
        j_l = 2
        j_r = len_w -1
        col_l = 'gram_' + str(j_l)
        col_m = 'gram_' + str(j_m)
        col_r = 'gram_' + str(j_r)
        val_l = dfw.iat[i, j_l+2]
        val_m = dfw.iat[i, j_m+2]
        val_r = dfw.iat[i, j_r+2]
        li_ind = df_.index[   (df_[col_r]==val_r) 
                            & (df_[col_m]==val_m)
                            & (df_[col_l]==val_l)
                            ]
        d_lilen[idx] = len(li_ind)

    # The dataframe must be returned - otherwise it will not be concatenated after parallelization 
    return dfw

While the processes work on different segments of our input dataframe we write results to a Python dictionary "d_lilen" which is under the control of the "parallelization manager" (see below). A dictionary is appropriate as we might otherwise loose control over the dataframe-indices during the following processes.

A reduced dataframe containing randomly selected "tokens"

To make things a bit easier we first create a "token"-dataframe "dfw_shorter3" based on a random selection of 100,000 indices from a dataframe containing long vocabulary words (length ≥ 10). We can derive it from our reference vocabulary. I have called the latter dataframe "dfw_short3" in the last post (because we use three 3-char-grams for longer tokens). "dfw_short3" contains all words of our vocabulary with a length of "10 ≤ length ≤ 30".

# Prepare a sub-dataframe for of the random 100,000 words 
# ******************************
num_w = 100000
len_dfw = len(dfw_short3)

# select a 100,000 random rows 
random.seed()
# Note: random.sample does not repeat values 
li_ind_p_w = random.sample(range(0, len_dfw), num_w)
len_li_p_w = len(li_ind_p_w)

dfw_shorter3 = dfw_short3.iloc[li_ind_p_w, :].copy() 
dfw_shorter3['lx'] = 0
dfw_shorter3['idx'] = dfw_shorter3.index
dfw_shorter3.head(5)

The resulting dataframe "dfw_shorter3" looks like :


You see that the index varies randomly and is not in ascending order! This is the reason why we must pick up the index-information during our parallelized operations!

Code for executing parallelized run

The following code enforces a parallelized execution:

manager = multiprocessing.Manager()
d_lilen = manager.dict()
print(len(d_lilen))

v_start_time = time.perf_counter()
dfw_res = parallelize(dfw_shorter3, getlen)
v_end_time = time.perf_counter()
cpu_time   = v_end_time - v_start_time
print("cpu : ", cpu_time)

print(len(d_lilen))
mean_length  = sum(d_lilen.values()) / len(d_lilen)
print(mean_length)

The parallelized run takes about 29.5 seconds.

cpu :  29.46206265499968
100000
1.25008

How does cpu-time vary with the number of cores of my (hyperthreading) CPU?

The cpu-time does not improve much when the number of cores gets bigger than the number of real physical cores:

1 core : 90.5 secs       
2 cores: 47.6 secs  
3 cores: 35.1 secs 
4 cores: 29.4 secs 
5 cores: 28.2 secs 
6 cores: 26.9 secs 
7 cores: 26.0 secs 
8 cores: 25.5 secs

My readers know about this effect already from ML experiments with CUDA and libcublas:

As long a s we use physical processor cores we see substantial improvement, beyond that no real gain in performance is observed on hyperthreading CPUs.

Compared to a run with just one CPU core we seem to gain a factor of almost 3 by parallelization. But, actually, this is no fair comparison: My readers have certainly seen that the CPU-time for the run with one CPU-Core is significantly slower than comparable runs which I described in my last post. At that time we found a cpu-time of around 75 secs, only. So, we have a basic deficit of about 15 secs - without real parallelization!

Overhead and RAM consumption of multiprocessing

Why does run with just one CPU core take so long time? Is it functional overhead for organizing and controlling multiprocessing - which may occur despite using just one core and just one "partition" of the dataframe (i.e. the full dataframe)? Well, we can test this easily by reconstructing the runs of my last post a bit:

# Reformulate Run just for cpu-time comparisons 
# **********************************************
b_test = True 

# Function  
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
def getleng(dfw, d_lileng):
    # Note 1: The dfw passed is a segment of the original dataframe  
    # Note 2: We use a list l_lilen which was outside defined 
    #         and is under the control of the prallelization manager
    
    num_rows = len(dfw)
    #print(num_rows)
    for i in range(0, num_rows):
        len_w = dfw.iat[i,0]
        idx = dfw.iat[i,33]
        
        df_name = "df_" + str(len_w)
        df_ = d_df[df_name]

        j_m = math.floor(len_w/2)+1
        j_l = 2
        j_r = len_w -1
        col_l = 'gram_' + str(j_l)
        col_m = 'gram_' + str(j_m)
        col_r = 'gram_' + str(j_r)
        val_l = dfw.iat[i, j_l+2]
        val_m = dfw.iat[i, j_m+2]
        val_r = dfw.iat[i, j_r+2]
        li_ind = df_.index[   (df_[col_r]==val_r) 
                            & (df_[col_m]==val_m)
                            & (df_[col_l]==val_l)
                            ]
        leng = len(li_ind)
        d_lileng[idx] = leng

    return d_lileng


if b_test: 
    num_w = 100000
    len_dfw = len(dfw_short3)

    # select a 100,000 random rows 
    random.seed()
    # Note: random.sample does not repeat values 
    li_ind_p_w = random.sample(range(0, len_dfw), num_w)
    len_li_p_w = len(li_ind_p_w)

    dfw_shortx = dfw_short3.iloc[li_ind_p_w, :].copy() 
    dfw_shortx['lx']  = 0
    dfw_shortx['idx'] = dfw_shortx.index

    d_lileng = {} #

    v_start_time = time.perf_counter()
    d_lileng = getleng(dfw_shortx, d_lileng)
    v_end_time = time.perf_counter()
    cpu_time   = v_end_time - v_start_time
    print("cpu : ", cpu_time)
    print(len(d_lileng))
    mean_length = sum(d_lileng.values()) / len(d_lileng)
    print(mean_length)
    
    dfw_shortx.head(3)

 
How long does such a run take?

cpu :  77.96989408900026
100000
1.25666

Just 78 secs! This is pretty close to the number of 75 secs we got in our last post's efforts! So, we see that turning to multiprocessing leads to significant functional overhead! The gain in performance, therefore, is less than the factor 3 observed above:

We (only) get a gain in performance by a factor of roughly 2.5 - when using 4 physical CPU cores.

I admit that I have no broad or detailed experience with Python multiprocessing. So, if somebody sees a problem in my code, please, send me a mail.

RAM is not released completely
Another negative side effect was the use of RAM in my case. Whereas we just get 2.2 GB RAM consumption with all required steps and copying parts of the loaded dataframe with all 3-char-grams in the above test run without multiprocessing, I saw a monstrous rise in memory during the parallelized runs:

Starting from a level of 2.4 GB, memory rose to 12.5 GB during the run and then fell back to 4.5 GB. So, there are copying processes and memory is not completely released again in the end - despite having all and everything encapsulated in functions. Repeating the multiprocessed runs even lead to a systematic increase in memory by about 150 MB per run.

So, when working with the "multiprocessing module" and big Pandas dataframes you should be a bit careful about the actual RAM consumption during the runs.

Conclusion

This series about finding words in a vocabulary by using two or three 3-char-grams may have appeared a bit "academical" - as one of my readers told me. Why the hell should someone use only a few 3-char-grams to identify words?

Well, I have tried to give some answers to this question: Under certain conditions you may only have fragments of words available; think of text transcribed from a recorded, but distorted communication with Skype or think of physically damaged written text documents. A similar situation may occur when you cannot trust a written string token to be a correctly written word - due to misspelling or other reasons (bad OCR SW or bad document conditions for scans combined with OCR).

In addition: character-grams are actually used as a basis for multiple ML methods for text-analysis tasks, e.g. in Facebook's Fasttext. They give a solid base for an embedded word vector space which can help to find and measure similarities between correctly written words, but also between correctly written words and fantasy words or misspelled words. Looking a bit at the question of how much a few 3-char-grams help to identify a word is helpful to understand their power in other contexts, too.

We have seen that only three 3-char-grams can identify matching words quite well - even if the words are long words (up to 30 characters). The list of matching words can be kept surprisingly small if and when

  • we use available or reasonable length information about the words we want to find,
  • we define positions for the 3-char-grams inside the words,
  • we put some positional distance between the location of the chosen 3-char-grams inside the words.

For a 100,000 random cases with correctly written 3-char-grams the average length of the hit list was below 2 - if the distance between the 3-char-grams was reasonably large compared to the token-length. Similar results were found for using only two 3-char-grams for short words.

We have also covered some very practical aspects regarding search operation on relatively big Pandas dataframes :

The CPU-time for identifying words in a Pandas dataframe by using 3-char-grams is reasonably small to allow for experiments with around 100,000 tokens even on PCs within minutes or quarters of an hour - but it does not take hours. As using 3-char-grams corresponds to putting conditions on two or three columns of a dataframe this result can be generalized to other similar problems with string comparisons on dataframe columns.

The basic RAM consumption of dataframes containing up to fifty-five 3-char-grams per word can be efficiently controlled by using the dtype "category" for the respective columns.

Regarding cpu-time we saw that working with many searches may get a performance boost by a factor well above 2 by using simple multiprocessing techniques based on Python's "multiprocessing" module. However, this comes with an unpleasant side effect of enormous RAM consumption - at least temporarily.

I hope you had some fun with this series of posts. In a forthcoming series I will apply these results to the task of error correction. Stay tuned.

Links

https://towardsdatascience.com/staying-sane-while-adopting-pandas-categorical-datatypes-78dbd19dcd8a
https://thispointer.com/python-pandas-select-rows-in-dataframe-by-conditions-

 

Pandas dataframe, German vocabulary – select words by matching a few 3-char-grams – II

In my last post

Pandas dataframe, German vocabulary – select words by matching a few 3-char-grams – I

I have discussed some properties of 3-char-grams of words in a German word list. (See the named post for the related structure of a Pandas dataframe ("dfw_uml") which hosts both the word list and all corresponding 3-char-grams.) In particular I presented the distribution of the maximum and mean number of words per unique 3-char-gram against the position of the 3-char-grams inside the the words of my vocabulary.

In the present post I want to use the very same Pandas dataframe to find German words which match two or three 3-char-grams defined at different positions inside some given strings or "tokens" of a text to be analyzed by a computer. One question in such a context is: How do we choose the 3-char-gram-positions to make the selection process effective in the sense of a short list of possible hits?

The dataframe has in my case 2.7 million rows for individual words and up to 55 columns for the values 3-char-grams at 55 positions. In the case of short words the columns are filled by artificial 3-char-grams "###".

My objective and a naive approach

Let us assume that we have a string (or "token") of e.g. 15 characters for a (German) word. The token contains some error in the sense of a wrongly written or omitted letter. Unfortunately, our text-analysis program does not know which letter of the string is wrongly written. So it wants to find words which may fit to the general character structure. We therefore pick a few 3-grams at given positions of our token. We then want to find words which match two or three 3-char-grams at different positions of the string - hoping that we chose 3-char-grams which do not contain any error. If we get no match we try different a different combination of 3-gram-positions.

In such a brute-force comparison process you would like to quickly pin down the number of matching words with a very limited bunch of 3-grams of the test token. The grams' positions should be chosen such that the hit list contains a minimum of fitting words. We, therefore, can pose this problem in a different way:

Which chosen positions or positional distances of two or three 3-char-grams inside a string token reduces the list of matching words from a vocabulary to a minimum?

Maybe there is a theoretically well founded solution for this problem. Personally, I am too old and too lazy to analyze such problems with solid mathematical statistics. I take a shortcut and trust my guts. It seems reasonable to me that the selected 3-char-grams should be distributed across the test string with a maximum distance between them. Let us see how far we get with this naive approach.

For the experiments discussed below I use

  • three 3-char-grams for tokens longer than 9 characters.
  • two 3-char-grams for tokens shorter than 9 letters.

For our first tests we pick correctly written 3-char-grams of test words. This means that we take correctly written words as our test tokens. The handling of tokens with wrongly written characters will be the topic of future articles.

Position combinations of two 3-char-grams for relatively short words

To get some idea about the problem's structure I first pick a test-word like "eisenbahn". As it is a relatively short word we start working with only two 3-char-grams. My test-word is an interesting one as it is a compound of two individual words "eisen" and "bahn". There are many other words in the German language which either contain the first or the second word. And in German we can add even more words to get even longer compounds. So, we would guess with some confidence that there are many hits if we chose two 3-char-grams overlapping each other or being located too close to each other. In addition we would also expect that we should use the length information about the token (or the sought words) during the selection process.

With a stride of 1 we have exactly seven 3-char-grams which reside completely inside our test-word. This gives us 21 options to use two 3-char-grams to find matching words.

To raise the chance for a bunch of alternative results we first look at words with up to 12 characters in our vocabulary and create a respective shortened slice of our dataframe "dfw_uml":

# Reduce the vocab to strings < max_len => Build dfw_short
#*********************************
#b_exact_length = False
b_exact_length = True

min_len = 4
max_len = 12
length  = 9

mil = min_len - 1 
mal = max_len + 1

if b_exact_length: 
    dfw_short = dfw_uml.loc[(dfw_uml.lower.str.len() == length)]
else:     
    dfw_short = dfw_uml.loc[(dfw_uml.lower.str.len() > mil) & (dfw_uml.lower.str.len() < mal)]
dfw_short = dfw_short.iloc[:, 2:26]
print(len(dfw_short))
dfw_short.head(5)

The above code allows us to choose whether we shorten the vocabulary to words with a length inside an interval or to words with a defined exact length. A quick and dirty code fragment to evaluate some statistics for all possible 21 position combinations for two 3-char-grams is the following:

# Hits for two 3-grams distributed over 9-letter and shorter words
# *****************************************************************
b_full_vocab  = False # operate on the full vocabulary 
#b_full_vocab  = True # operate on the full vocabulary 

word  = "eisenbahn"
word  = "löwenzahn"
word  = "kellertür"
word  = "nashorn"
word  = "vogelart"

d_col = { "col_0": "gram_2", "col_1": "gram_3", "col_2": "gram_4", "col_3": "gram_5",
          "col_4": "gram_6", "col_5": "gram_7", "col_6": "gram_8" 
        }
d_val = {}
for i in range(0,7):
    key_val  = "val_" + str(i)
    sl_start = i
    sl_stop  = sl_start + 3
    val = word[sl_start:sl_stop] 
    d_val[key_val] = val
print(d_val)

li_cols = [0] # list of cols to display in a final dataframe 

d_num = {}
 words 
# find matching words for all position combinations
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
upper_num = len(word) - 2 
for i in range(0,upper_num): 
    col_name1 = "col_" + str(i)
    val_name1 = "val_"  + str(i)
    col1 = d_col[col_name1]
    val1 = d_val[val_name1]
    col_name2 = ''
    val_name2 = ''
    for j in range(0,upper_num):
        if j <= i : 
            continue 
        else:
            col_name2 = "col_" + str(j)
            val_name2 = "val_"  + str(j)
            col2 = d_col[col_name2]
            val2 = d_val[val_name2]
            
            # matches ?
            if b_full_vocab:
                li_ind = dfw_uml.index[  (dfw_uml[col1]==val1) 
                                    &    (dfw_uml[col2]==val2)
                                      ].tolist()
            else: 
                li_ind = dfw_short.index[(dfw_short[col1]==val1) 
                                    &    (dfw_short[col2]==val2)
                                        ].tolist()
                
            num = len(li_ind)
            key = str(i)+':'+str(j)
            d_num[key] = num
#print("length of d_num = ", len(d_num))
print(d_num)

# bar diagram 
fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 6
names  = list(d_num.keys())
values = list(d_num.values())
plt.bar(range(len(d_num)), values, tick_label=names)
plt.xlabel("positions of the chosen two 3-grams", fontsize=14, labelpad=18)
plt.ylabel("number of matching words", fontsize=14, labelpad=18)
font_weight = 'bold' 
font_weight = 'normal' 
if b_full_vocab: 
    add_title = "\n(full vocabulary)"
elif  (not b_full_vocab and not b_exact_length):
    add_title = "\n(reduced vocabulary)"
else:
    add_title = "\n(only words with length = 9)"
    
plt.title("Number of words for different position combinations of two 3-char-grams" + add_title, 
          fontsize=16, fontweight=font_weight, pad=18) 
plt.show()

 
You see that I prepared three different 9-letter words. And we can choose whether we want to find matching words of the full or of the shortened dataframe.

The code, of course, imposes conditions on two columns of the dataframe. As we are only interested in the number of resulting words we use these conditions together with the "index()"-function of Pandas.

Number of matching relatively short words against position combinations for two 3-char-grams

For the full vocabulary we get the following statistics for the test-word "eisenbahn":

{'val_0': 'eis', 'val_1': 'ise', 'val_2': 'sen', 'val_3': 'enb', 'val_4': 'nba', 'val_5': 'bah', 'val_6': 'ahn'}
{'0:1': 5938, '0:2': 5899, '0:3': 2910, '0:4': 2570, '0:5': 2494, '0:6': 2500, '1:2': 5901, '1:3': 2910, '1:4': 2570, '1:5': 2494, '1:6': 2500, '2:3': 3465, '2:4': 2683, '2:5': 2498, '2:6': 2509, '3:4': 4326, '3:5': 2681, '3:6': 2678, '4:5': 2836, '4:6': 2832, '5:6': 3857}

Note: The first and leftmost 3-char-gram is located at position "0", i.e. we count positions from zero. Then the last position is at position "word-length - 3".

The absolute numbers are much too big. But this plot already gives a clear indication that larger distances between the two 3-char-grams are better to limit the size of the result set. When we use the reduced vocabulary slice (with words shorter than 13 letters) we get

{'0:1': 1305, '0:2': 1277, '0:3': 143, '0:4': 48, '0:5': 20, '0:6': 24, '1:2': 1279, '1:3': 143, '1:4': 48, '1:5': 20, '1:6': 24, '2:3': 450, '2:4': 125, '2:5': 23, '2:6': 31, '3:4': 634, '3:5': 58, '3:6': 55, '4:5': 76, '4:6': 72, '5:6': 263}

For some combinations the resulting hit list is much shorter (< 50)! And the effect of some distance between the chosen char-grams gets much more pronounced.

Corresponding data for the words "löwenzahn" and "kellertür" confirm the tendency:

Test-word "löwenzahn"

Watch the lower numbers along the y-scale!

Test-token "kellertür"

Using the information about the word length for optimization

On average the above numbers are still too big for a later detailed comparative analysis with our test token - even on the reduced vocabulary. We expect an improvement by including the length information. What numbers do we get when we use a list with words having exactly the same length as the test-word?

You find the results below:

Test-token "eisenbahn"

{'0:1': 158, '0:2': 155, '0:3': 16, '0:4': 6, '0:5': 1, '0:6': 3, '1:2': 155, '1:3': 16, '1:4': 6, '1:5': 1, '1:6': 3, '2:3': 83, '2:4': 37, '2:5': 3, '2:6': 9, '3:4': 182, '3:5': 17, '3:6': 17, '4:5': 22, '4:6': 22, '5:6': 109}

Test-token "löwenzahn"

{'0:1': 94, '0:2': 94, '0:3': 3, '0:4': 2, '0:5': 2, '0:6': 1, '1:2': 94, '1:3': 3, '1:4': 2, '1:5': 2, '1:6': 1, '2:3': 3, '2:4': 2, '2:5': 2, '2:6': 1, '3:4': 54, '3:5': 43, '3:6': 13, '4:5': 59, '4:6': 14, '5:6': 46}

Test-token "kellertür"

{'0:1': 14, '0:2': 13, '0:3': 13, '0:4': 5, '0:5': 1, '0:6': 1, '1:2': 61, '1:3': 24, '1:4': 5, '1:5': 1, '1:6': 2, '2:3': 36, '2:4': 8, '2:5': 1, '2:6': 3, '3:4': 12, '3:5': 1, '3:6': 1, '4:5': 17, '4:6': 17, '5:6': 17}

For an even shorter word like "vogelart" and "nashorn" two 3-char-grams cover almost all of the word. But even here the number of hits is largest for neighboring 3-char-grams:

Test-word "vogelart" (8 letters)

{'val_0': 'vog', 'val_1': 'oge', 'val_2': 'gel', 'val_3': 'ela', 'val_4': 'lar', 'val_5': 'art', 'val_6': 'rt'}
{'0:1': 22, '0:2': 22, '0:3': 1, '0:4': 1, '0:5': 1, '1:2': 23, '1:3': 1, '1:4': 1, '1:5': 2, '2:3': 10, '2:4': 6, '2:5': 5, '3:4': 19, '3:5': 15, '4:5': 24}

Test-word "nashorn" (7 letters)

{'val_0': 'nas', 'val_1': 'ash', 'val_2': 'sho', 'val_3': 'hor', 'val_4': 'orn', 'val_5': 'rn', 'val_6': 'n'}
{'0:1': 1, '0:2': 1, '0:3': 1, '0:4': 1, '1:2': 1, '1:3': 1, '1:4': 1, '2:3': 3, '2:4': 2, '3:4': 26}

So, as an intermediate result I would say:

  • Our naive idea about using 3-char-grams with some distance between them is pretty well confirmed for relatively small words with a length below 9 letters and two 3-char-grams.
  • We should use the length information about a test-word or token in addition to diminish the list of reasonably matching words!

Code to investigate 3-char-gram combinations for words with more than 9 letters

Let us now turn to longer words. Here we face a problem: The number of possibilities to choose three 3-char-grams at different positions explodes with word-length (simple combinatorics leading to the binomial coefficient). It is even difficult to present results graphically. Therefore, I had to restrict myself to gram-combinations with some reasonable distance from the beginning.

The following code does not exclude anything and leads to problematic plots:

# Hits for two 3-grams distributed over a 13-letter word
# ******************************************************
b_full_vocab  = False # operate on the full vocabulary 
#b_full_vocab  = True # operate on the full vocabulary 

#word  = "nachtwache"             # 10
#word  = "morgennebel"            # 11
#word  = "generalmajor"           # 12
#word  = "gebirgskette"           # 12
#word  = "fussballfans"           # 12
#word  = "naturforscher"          # 13
#word  = "frühjahrsputz"          # 13 
#word  = "marinetaucher"          # 13
#word  = "autobahnkreuz"          # 13 
word  = "generaldebatte"         # 14
#word  = "eiskunstläufer"         # 14
#word  = "gastwirtschaft"         # 14
#word  = "vergnügungspark"        # 15 
#word  = "zauberkuenstler"        # 15
#word  = "abfallentsorgung"       # 16 
#word  = "musikveranstaltung"     # 18  
#word  = "sicherheitsexperte"     # 18
#word  = "literaturwissenschaft"  # 21 
#word  = "veranstaltungskalender" # 23

len_w = len(word)
print(len_w, math.floor(len_w/2))

d_col = { "col_0": "gram_2",   "col_1": "gram_3",   "col_2": "gram_4",   "col_3": "gram_5",
          "col_4": "gram_6",   "col_5": "gram_7",   "col_6": "gram_8",   "col_7": "gram_9", 
          "col_8": "gram_10",  "col_9": "gram_11",  "col_10": "gram_12", "col_11": "gram_13", 
          "col_12": "gram_14", "col_13": "gram_15", "col_14": "gram_16", "col_15": "gram_17", 
          "col_16": "gram_18", "col_17": "gram_19", "col_18": "gram_20", "col_19": "gram_21" 
        }
d_val = {}

ind_max = len_w - 2

for i in range(0,ind_max):
    key_val  = "val_" + str(i)
    sl_start = i
    sl_stop  = sl_start + 3
    val = word[sl_start:sl_stop] 
    d_val[key_val] = val
print(d_val)

li_cols = [0] # list of cols to display in a final dataframe 

d_num = {}
li_permut = []

# prepare short
length  = len_w
mil = min_len - 1 
mal = max_len + 1
b_exact_length = True
if b_exact_length: 
    dfw_short = dfw_uml.loc[(dfw_uml.lower.str.len() == length)]
else:     
    dfw_short = dfw_uml.loc[(dfw_uml.lower.str.len() > mil) & (dfw_uml.lower.str.len() < mal)]
dfw_short = dfw_short.iloc[:, 2:26]
print(len(dfw_short))


# find matching words for all position combinations
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
for i in range(0,ind_max): 
    for j in range(0,ind_max):
        for k in range(0,ind_max):
            if (i,j,k) in li_permut or (i==j or j==k or i==k):
                continue
            else: 
                col_name1 = "col_" + str(i)
                val_name1 = "val_" + str(i)
                col1 = d_col[col_name1]
                val1 = d_val[val_name1]
                col_name2 = "col_" + str(j)
                val_name2 = "val_" + str(j)
                col2 = d_col[col_name2]
                val2 = d_val[val_name2]
                col_name3 = "col_" + str(k)
                val_name3 = "val_" + str(k)
                col3 = d_col[col_name3]
                val3 = d_val[val_name3]
                li_akt_permut = list(itertools.permutations([i, j, k]))
                li_permut = li_permut + li_akt_permut
                #print("i,j,k = ", i, ":", j, ":", k)
                #print(len(li_permut))
                
                # matches ?
                if b_full_vocab:
                    li_ind = dfw_uml.index[  (dfw_uml[col1]==val1) 
                                        &    (dfw_uml[col2]==val2)
                                        &    (dfw_uml[col3]==val3)
                                          ].tolist()
                else: 
                    li_ind = dfw_short.index[(dfw_short[col1]==val1) 
                                        &    (dfw_short[col2]==val2)
                                        &    (dfw_short[col3]==val3)
                                            ].tolist()

                num = len(li_ind)
                key = str(i)+':'+str(j)+':'+str(k)
                d_num[key] = num
print("length of d_num = ", len(d_num))
print(d_num)

# bar diagram 
fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 15
fig_size[1] = 6
names  = list(d_num.keys())
values = list(d_num.values())
plt.bar(range(len(d_num)), values, tick_label=names)
plt.xlabel("positions of the chosen two 3-grams", fontsize=14, labelpad=18)
plt.ylabel("number of matching words", fontsize=14, labelpad=18)
font_weight = 'bold' 
font_weight = 'normal' 
if b_full_vocab: 
    add_title = "\n(full vocabulary)"
elif  (not b_full_vocab and not b_exact_length):
    add_title = "\n(reduced vocabulary)"
else:
    add_title = "\n(only words with length = " + str(len_w) + ")"
    
plt.title("Number of words for different position combinations of two 3-char-grams" + add_title, 
          fontsize=16, fontweight=font_weight, pad=18) 
plt.show()

 

An example for the word "generaldebatte" (14 letters) gives:

A supplemental code that reduces the set of gram position combinations significantly to larger distances could look like this:

# Analysis for 3-char-gram combinations with larger positional distance
# ********************************************************************

hf = math.floor(len_w/2)

d_l={}
for i in range (2,26):
    d_l[i] = {}

for key, value in d_num.items():
    li_key = key.split(':')
    # print(len(li_key))
    i = int(li_key[0])
    j = int(li_key[1])
    k = int(li_key[2])
    l1 = int(li_key[1]) - int(li_key[0])
    l2 = int(li_key[2]) - int(li_key[1])
    le = l1 + l2 
    # print(le)
    if (len_w < 12): 
        bed1 = (l1<=1 or l2<=1)
        bed2 = (l1 <=2 or l2 <=2)
        bed3 = (((i < hf and j< hf and k< hf) or (i > hf and j> hf and k > hf)))
    if (len_w < 15): 
        bed1 = (l1<=2 or l2<=2)
        bed2 = (l1 <=3 or l2 <=3)
        bed3 = (((i < hf and j< hf and k< hf) or (i > hf and j> hf and k > hf)))
    elif (len_w <18): 
        bed1 = (l1<=3 or l2<=3)
        bed2 = (l1 <=4 or l2 <=4)
        bed3 = (((i < hf and j< hf and k< hf) or (i > hf and j> hf and k > hf)))
    else: 
        bed1 = (l1<=3 or l2<=3)
        bed2 = (l1 <=4 or l2 <=4)
        bed3 = (((i < hf and j< hf and k< hf) or (i > hf and j> hf and k > hf)))
        
    for j in range(2,26): 
        if le == j:
            if value == 0 or bed1 or ( bed2 and bed3) : 
                continue
            else:
                d_l[j][key] = value

sum_len = 0 
n_p = len_w -2
for j in range(2,n_p):
    num = len(d_l[j])
    print("len = ", j, " : ", "num = ", num) 
    
print()
print("len_w = ", len_w, " half = ", hf)    

if (len_w <= 12):
    p_start = hf 
elif (len_w < 15):
    p_start = hf + 1
elif len_w < 18: 
    p_start = hf + 2 
else: 
    p_start = hf + 2 

    
# Plotting 
# ***********
li_axa = []
m = 0
for i in range(p_start,n_p):
    if len(d_l[i]) == 0:
        continue
    else:
        m+=1
print(m)
fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = m * 5
fig_b  = plt.figure(2)

for j in range(0, m):
    li_axa.append(fig_b.add_subplot(m,1,j+1))

m = 0
for i in range(p_start,n_p):
    if len(d_l[i]) == 0:
        continue
    # bar diagram 
    names  = list(d_l[i].keys())
    values = list(d_l[i].values())
    li_axa[m].bar(range(len(d_l[i])), values, tick_label=names)
    li_axa[m].set_xlabel("positions of the 3-grams", fontsize=14, labelpad=12) 
    li_axa[m].set_ylabel("num matching words", fontsize=14, labelpad=12) 
    li_axa[m].set_xticklabels(names, fontsize=12, rotation='vertical')
    #font_weight = 'bold' 
    font_weight = 'normal' 
    if b_full_vocab: 
        add_title = " (full vocabulary)"
    elif  (not b_full_vocab and not b_exact_length):
        add_title = " (reduced vocabulary)"against position-combinations for <em>three</em> 3-char-grams</h1>
    else:
        add_title = " (word length = " + str(len_w) + ")" 

    li_axa[m].set_title("total distance = " + str(i) + add_title, 
              fontsize=16, fontweight=font_weight, pad=16) 
    m += 1
    
plt.subplots_adjust( hspace=0.7 )
fig_b.suptitle("word :  " + word +" (" + str(len_w) +")", fontsize=24, 
              fontweight='bold', y=0.91) 
plt.show()

 

What are the restrictions? Basically

  • we eliminate combinations with 2 neighboring 3-char-grams,
  • we eliminate 3-char-grams combinations where all 3-grams are place only on one side of the word - the left or right one,
  • we pick only 3-char-grams where the sum of the positional distances between the 3-char-grams is somewat longer than half of the token's length.

We vary these criteria a bit with the word length. In my opinion these criteria should produce plots, only, which show that the number of hits is reasonably small - if our basic approach is of some value.

Number of matching words with more than 9 letters against position-combinations for three 3-char-grams

The following plots cover words of different growing lengths for dataframes reduced to words with exactly the same length as the chosen token. Not too surprising, all of the words are compound words.

**************************

Test-token "nachtwache"

Test-token "morgennebel"

Test-token "generalmajor"

Test-token "gebirgskette"

Test-token "fussballfans"

Test-token "naturforscher"

Test-token "frühjahrsputz"

Test-token "marinetaucher"

Test-token "autobahnkreuz"

Test-token "generaldebatte"

Test-token "eiskunstläufer"

Test-token "gastwirtschaft"

Test-token "vergnügungspark"

Test-token "zauberkuenstler"

Test-token "abfallentsorgung"

Test-token "musikveranstaltung"

Test-token "sicherheitsexperte"

Test-token "literaturwissenschaft"

Test-token "veranstaltungskalender"

**************************

What we see is that whenever we choose 3-char-gram combinations with a relative big positional distance between them and a sum of the two distances ≥ word-length / 2 + 2 the number of matching words ogf the vocabulary is smaller than 10, very often even smaller than 5. The examples "prove" at least that choosing three (correctly written) 3-char-grams with relative big distance within a token lead to small numbers of matching vocabulary words,

Conclusion

One can use a few 3-char-grams within string tokens to find matching vocabulary words via a comparison of the char-grams at their respective position. In this article we have studied how we should choose two or three 3-char-grams within string tokens of length ≤ 9 letters or > 9 letters, respectively, if and when we want to find matching vocabulary words effectively. We found strong indications that the 3-char-grams should be chosen with a relatively big positional distance. To use neighboring 3-char-grams will lead to hit numbers which are too big for a detailed analysis.

In the next post I will have a closer look at the required CPU-time for a word searches in a vocabulary based on 3-char-gram comparisons for a 100,000 string tokens.

 
 

Pandas dataframe, German vocabulary – select words by matching a few 3-char-grams – I

Words or strings can be segmented into so called "n-character-grams" or "n-char-grams". A n-char-gram is a defined sequence of "n" letters, i.e. a special string of length "n". Such a defined letter sequence - if short enough - can be found at various positions within many words of a vocabulary. Words or technically speaking "strings" can e.g. be thought of being composed of a sequence of defined "2-char-grams" or "3-char-grams". "n-char-grams" are useful for text-analysis and/or machine-learning methods applied to texts.

Let us assume you have a string representing a test word - but unfortunately with one or two wrong characters or two transposed characters at certain positions inside it. You may nevertheless want to find words in a German vocabulary which match most of the correct letters. One naive approach could be to compare the characters of the string position-wise with corresponding characters of all words in the vocabulary and then pick the word with most matches. As you neither can trust the first character nor the last character you quickly understand that a quick and efficient way of raising the probability to find reasonable fitting words requires to compare not only single letters but also bunches of them, i.e. sub-strings of sequential letters or "n-char-grams".

This defines the problem of comparing n-char-grams at certain positions inside string "tokens" extracted from unknown texts with n-char-grams of words in a vocabulary. I call a "token" an unchecked distinct letter sequence, i.e. a string, identified by some "Tokenizer"-algorithm, which was applied to a text. A Tokenizer typically identifies word-separator characters to do his job. A "token" might or might not be regular word of a language.

This mini-series looks a bit at using "3-character-grams" of words in a German vocabulary residing in a Pandas dataframe. Providing and using 3-grams of a huge vocabulary in a suitable form as input for Python functions working on a Pandas dataframe can, however, be a costly business:

  • RAM: First of all Pandas dataframes containing strings in most of the columns require memory. Using the dtype "category" helps a lot to limit the memory consumption for a dataframe comprising all 3-char-grams of a reasonable vocabulary with some million words. See my last post on this topic.
  • CPU-time: Another critical aspect is the CPU-time required to determine all dataframe rows, i.e. vocabulary words, which contain some given 3-char-grams at defined positions.
  • It is not at all clear how many 3-char-grams are required to narrow down the spectrum of fitting words (of the vocabulary) for a given string to a small amount which can be handled by further detailed analysis modules.

In this article I, therefore, look at "queries" on a Pandas dataframe containing vocabulary words plus their 3-char-grams at defined positions inside the words. Each column contains 3-char-grams at a defined position in the word strings. Our queries apply conditions to multiple selected columns. I first discuss how 3-char-grams split the vocabulary into groups. I present some graphs of how the number of words for such 3-char-gram based groups vary with 3-gram-position. Then the question how many 3-char-grams at different positions allow for an identification of a reasonably small bunch of fitting words in the vocabulary will be answered by some elementary experiments. We also look at CPU-times required for related queries and I discuss some elementary optimization steps. An eventual short turn to multiprocessing reveals that we, indeed, can gain a bit of performance.

As a basis for my investigations I use a "vocabulary" based on the work of Torsten Brischalle. See from http://www.aaabbb.de/WordList/WordList.php. I have supplemented his word-list by words with different writings for Umlauts. The word list contains around 2.8 million German words. Regarding the positional shift of the 3-char-grams of a word against each other I use the term "stride" as explained in my last post
Pandas and 3-char-grams of a vocabulary – reduce memory consumption by datatype „category“.
In addition I use some "padding" and fill up 3-char-grams at and beyond word boundaries with special characters (see the named post for it). In some plots I abbreviated "3-char-grams" to "3-grams".

Why do I care about CPU-time on Pandas dataframes with 3-char-grams?

CPU-time is important if you want to correct misspelled words in huge bunches of texts with the help of 3-char-gram segmentation. Misspelled words are not only the result of wrong writing, but also of bad scans of old and unclear texts. I have a collection of over 200,000 such scans of German texts. The application of the Keras Tokenizer produced around 1.9 million string tokens.

Around 50% of the most frequent 100.000 tokens in my scanned texts appear to have "errors" as they are no members of the (limited) vocabulary. The following plot shows the percentage of hits in the vocabulary against the absolute number of the most frequent words within the text collection:

The "errors" contain a variety of (partially legitimate) compound words outside the vocabulary, but there are also wrong letters at different positions and omitted letters due to a bad OCR-quality of the scans. Correcting at least some of the simple errors (as one or two wrong characters) could improve the quality of the scan results significantly. To perform an analysis based on 3-char-grams we have to compare tenths up to hundreds of thousands tokens with some million vocabulary words. CPU-time matters - especially when using Pandas as a kind of database.

As the capabilities of my Linux workstation are limited I was interested in whether an analysis of 100,000 misspelled words based on comparisons of 3-char-grams is within reach for lets say a 100,000 tokens on a reasonably equipped PC.

Major Objective: Reduce the amount of vocabulary words matching a few 3-char-grams at different string positions to a minimum

The analysis of possible errors of a scanned word is more difficult than one may think. The errors may be of different nature and may have different consequences for the length and structure of the resulting error-containing word in comparison with the originally intended word. Different error types may appear in combination and the consequences may interfere within a word (or identified token).

What you want to do is to find words in the vocabulary which are comparable to your token - at least in some major parts. The list of such words would be those which with some probability might contain the originally intended word. Then you might apply a detailed and error specific analysis to this bunch of interesting words. Such an analysis may be complemented by an additional analysis on (embedded) word-vector spaces created by ML-trained neural networks to predict words at the end of a sequence of other words. A detailed analysis on a list of words and their character composition in comparison to a token may be CU-time intensive in itself as it typically comprises string operations.

In addition it is required to do the job a bit differently for certain error types and you also have to make some assumptions regarding the error's impact on the word-length. But even under simplifying assumptions regarding the number of wrong letters and the correct total amount of letters in a token, you are confronted with a basic problem of error-correction:

You do not know where exactly a mistake may have occurred during scanning or wrong writing.

As a direct consequence you may have to compare 3-char-grams at various positions within the token with corresponding 3-char-grams of vocabulary words. But more queries mean more CPU-time ....

In any case one major objective must be to quickly reduce the amount of words of the vocabulary which you want to use in the detailed error analysis down to a minimum below 10 words with only a few Pandas queries. Therefore, two points are of interest here:

  • How does the number of 3-char-grams for vocabulary words vary with the position?
  • How many correct 3-char-grams define a word in the vocabulary on average?

The two aspects may, of course, be intertwined.

Structure of the Pandas dataframe containing the vocabulary and its 3-char-grams

The image below displays the basic structure of the vocabulary I use in a Pandas dataframe (called "dfw_uml"):

The column "len" contains the length of a word. The column "indw" is identical to "lower". "indw" allows for a quick change of the index from integers to the word itself. Each column with "3-char-gram" in the title corresponds to a defined position of 3-char-grams.

The stride between adjacent 3-char-grams is obviously 1. I used a "left-padding" of 2. This means that the first 3-char-grams were supplemented by the artificial letter "%" to the left. The first 3-char-gram with all letters residing within the word is called "gram_2" in my case - with its leftmost letter being at position 0 of the word-string and the rightmost letter at position 2. On the right-most side of the word we use the letter "#" to create 3-char-grams reaching outside the word boundary. You see that we get many "###" 3-char-grams for short words at the right side of the dataframe.

Below I actually use two dataframes: one with 21 3-char-grams up to position 21 and another one with (55) 3-char-grams up to position 55.

Variation of the number of vocabulary words against their length

With growing word-length there are more 3-char-grams to look at. Therefore we should have an idea about the distribution of the number of words with respect to word-length. The following plot shows how many different words we find with growing word-length in our vocabulary:

The Python code for the plot above is :

x1 = []
y1 = []
col_name = 'len'
df_col_grp_len = dfw_uml.groupby(col_name)['indw'].count()
d_len_voc = df_col_grp_len.to_dict()
#print (df_col_grp_len)
#print(d_len_voc) 

len_d = len(d_len_voc)
for key,value in d_len_voc.items():
    x1.append(key)
    y1.append(value)

fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 6    
plt.plot(x1,y1, color='darkgreen', linewidth=5)
#plt.xticks(x)
plt.xlabel("length of word)", fontsize=14, labelpad=18)
plt.ylabel("number of words ", fontsize=14, labelpad=18)
plt.title("Number of different words against length ") 
plt.show()

 

So, the word-length interval between 2 and 30 covers most of the words. This is consistent with the Pandas information provided by Pandas' "describe()"-function applied to column "len":

How does the number of different 3-char-grams vary with the 3-char-gram position?

Technically a 3-char-gram can be called "unique" if it has a specific letter-sequence at a specific defined position. So would call the 3-char-grams "ena" at position 5 and "ena" at position 12 unique despite their matching sequence of letters.

There is only a limited amount of different 3-char-gram at a given position within the words of a given vocabulary.
Each 3-char-gram column of our dataframe can thus be divided into multiple "categories" or groups of words containing the same specific 3-char-gram at the position associated with the column. A priori t was not at all clear to me how many vocabulary words we would typically find for a given 3-char-gram at a defined position. I wanted an overview. So let us first look at the number of different 3-char-grams against position.

So how does the distribution of the number of unique 3-char-grams against position look like?

To answer this question we use the Pandas function nunique() in the following way:

# Determine number of unique values in columns )(i.e. against 3-char-gram position)
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
unique_vals = dfw_uml.nunique()
luv = len(unique_vals)
print(unique_vals)

and get

.....
.....
gram_0          29
gram_1         459
gram_2        3068
gram_3        4797
gram_4        8076
gram_5        8687
gram_6        8743
gram_7        8839
gram_8        8732
gram_9        8625
gram_10       8544
gram_11       8249
gram_12       7829
gram_13       7465
gram_14       7047
gram_15       6700
gram_16       6292
gram_17       5821
gram_18       5413
gram_19       4944
gram_20       4452
gram_21       3989

Already in my last post we saw that the given different 3-char-grams at a defined position divide the vocabulary into a relatively small amount of groups. For my vocabulary with 2.8 million words the maximum number of different 3-char-grams is around 8,800 at position 7 (for a stride of 1). 8,800 is relatively small compared to the total number of 2.7 million words.

Above I looked at the 3-char-grams at the first 21 positions (including left-padding 3-char-grams). We can get a plot by applying the the following code

# Plot for the distribution of categories (i.e. different 3-char-grams) against position
# **************************************
li_x = []
li_y = []
sum = 0 

for i in range(0, luv-4):
    li_x.append(i)
    name = 'gram_' + str(i)
    n_diff_grams = unique_vals[name] 
    li_y.append(n_diff_grams)
    sum += n_diff_grams
print(sum)

fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 6
plt.plot(li_x,li_y, color='darkblue', linewidth=5)
plt.xlim(1, 22)
plt.xticks(li_x)
plt.xlabel("3-gram position (3rd character)", fontsize=14, labelpad=18)
plt.ylabel("number of different 3-grams", fontsize=14, labelpad=18)
plt.show()

The plot is:

We see a sharp rise of the number of different 3-char-grams with position 2 (i.e. with the 1st real character of the word) and a systematic decline after position 11. The total sum of all unique 3-char-grams over all positions 136,800 for positions up to 21. (The number includes padding-left and padding-right 3-char-grams).

When we extend the number of positions of 3-char-grams from 0 to 55 we get:

The total sum of unique 3-char-grams then becomes 161,259.

Maximum number of words per unique 3-char-gram with position

In a very similar way we can get the maximum number of rows, i.e. of different vocabulary words, appearing for a specific 3-char-gram at a certain position. This specific 3-char-gram defines the largest category or word group at the defined position. The following code creates a plot for the variation of this maximum against the 3-char-gram-position:

# Determine max number of different rows per category
# ***********************************************
x = []
y = []
i_min = 0; i_max = 56
for j in range(i_min, i_max):
    col_name = 'gram_' + str(j)
    maxel = dfw_uml.groupby(col_name)['indw'].count().max()
    x.append(j)
    y.append(maxel)

fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 6    
plt.plot(x,y, color='darkred', linewidth=5)
plt.xticks(x)
plt.xlabel("3-gram position (3rd character)", fontsize=14, labelpad=18)
plt.ylabel("max number of words per 3-gram", fontsize=14, labelpad=18)
plt.show()

The result is:

The fact that there are less and less words with growing length in the vocabulary explains the growing maximum number of words for 3-char-grams at a defined late position. The maximum there corresponds to words for the artificial 3-char-gram "###". Also the left-padding 3-char-grams have many fitting words.

Consistent to the number of different categories we get relatively small numbers between positions 3 and 9:

Note that above we looked at the maximum, only. The various 3-char-grams defined at a certain position may have very different numbers of words being consistent with the 3-char-gram.

Mean number of words with 3-char-gram position and variation at a certain position

Another view at the number of words per unique 3-char-gram is given by the average number of words for the 3-char-grams with position. The following graphs were produced by replacing the max()-function in the code above by the mean()-function:

Mean number of words per 3-char-gram category against positions 0 to 55:

Mean number of words per 3-char-gram category against positions 0 to 45:

We see that there is a significant slope after position 40. Going down to lower positions we see a more modest variation.

There is some variation, but the total numbers are much smaller than the maximum numbers. This means that there is only a relatively small number of 3-char-grams which produce real big numbers.

This can also be seen from the following plots where I have ordered the 3-char-grams according to the rising number of matching words for the 3-char-grams at position 5 and at position 10:

Watch the different y-scales! When we limit the number of ordered grams to 8000 the variation is much more comparable:

Conclusion

A quick overview over a vocabulary with the help of Pandas functions shows that the maximum and the mean number of matching words for 3-char-grams at defined positions inside the vocabulary words vary strongly with position and thereby also with word-length.

In the position range from 4 to 11 the mean number of words per unique 3-char-gram is pretty small - around 320. In the position range between 4 and 30 (covering most of the words) the mean number of different words per 3-char-gram is still below 1000.

This gives us some hope for reducing the number of words matching a few 3-char-grams at different positions down to numbers we can handle when applying a detailed analysis. The reason is that we then are interested in the intersection of multiple matching word-groups at the different positions. Respective queries, hit rates and CPU-Times are the topic of the next article:

Pandas dataframe, German vocabulary – select words by matching a few 3-char-grams – II

Stay tuned ...