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Python Notebook Download

Error Propagation - Liq Only

• This workbook demonstrates how to investigate the effect of uncertainty on calculated temperatures for liquid-only thermometers

• You should use this as a nice simple example to understand how Thermobar can add noise

• You can download the required excel spreadsheet at https://github.com/PennyWieser/Thermobar/blob/main/docs/Examples/Error_propagation/Liquid_Errors.xlsx

You need to install Thermobar once on your machine, if you haven’t done this yet, uncomment the line below (remove the #)

#!pip install Thermobar

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import Thermobar as pt
pd.options.display.max_columns = None

This sets plotting parameters

# This sets some plotting things
plt.rcParams["font.family"] = 'arial'
plt.rcParams["font.size"] =12
plt.rcParams["mathtext.default"] = "regular"
plt.rcParams["mathtext.fontset"] = "dejavusans"
plt.rcParams['patch.linewidth'] = 1
plt.rcParams['axes.linewidth'] = 1
plt.rcParams["xtick.direction"] = "in"
plt.rcParams["ytick.direction"] = "in"
plt.rcParams["ytick.direction"] = "in"
plt.rcParams["xtick.major.size"] = 6 # Sets length of ticks
plt.rcParams["ytick.major.size"] = 4 # Sets length of ticks
plt.rcParams["ytick.labelsize"] = 12 # Sets size of numbers on tick marks
plt.rcParams["xtick.labelsize"] = 12 # Sets size of numbers on tick marks
plt.rcParams["axes.titlesize"] = 14 # Overall title
plt.rcParams["axes.labelsize"] = 14 # Axes labels

Example 1: Absolute errors in wt%

• input spreadsheet has absolute errors (in wt%) as column headings (e.g., from experimental studies where they report 1 sigma uncertainties)

• We want to generate N synthetic liquids for each measured liquid whose parameters vary within these error bounds.

# this cell loads the oxide data, e.g., colum headings SiO2_Liq, MgO_Liq etc.
out=pt.import_excel('Liquid_Errors.xlsx', sheet_name="Error_Example_Abs")
my_input=out['my_input']
myLiquids1=out['Liqs']

# This cell loads the errors, reading from columns SiO2_Liq_Err, MgO_Liq_Err etc.
out_Err=pt.import_excel_errors('Liquid_Errors.xlsx', sheet_name="Error_Example_Abs")
myLiquids1_Err=out_Err['Liqs_Err']
myinput_Out=out_Err['my_input_Err']

# Check everything has read in correctly
display(myLiquids1_Err.head())
display(myLiquids1.head())

	SiO2_Liq_Err	TiO2_Liq_Err	Al2O3_Liq_Err	FeOt_Liq_Err	MnO_Liq_Err	MgO_Liq_Err	CaO_Liq_Err	Na2O_Liq_Err	K2O_Liq_Err	Cr2O3_Liq_Err	P2O5_Liq_Err	H2O_Liq_Err	Fe3Fet_Liq_Err	NiO_Liq_Err	CoO_Liq_Err	CO2_Liq_Err	Sample_ID_Liq_Err	P_kbar_Err	T_K_Err
0	0.168800	0.094674	0.352282	0.185453	0.003443	0.423718	0.185089	0.424152	0.273896	0.007649	0.011024	0.2	0.0	0.0	0.0	0.0	0	0.1	5
1	0.273773	0.043005	0.128686	0.348039	0.062106	0.422937	0.427487	0.255009	0.455046	0.005722	0.032563	0.2	0.0	0.0	0.0	0.0	1	0.1	5
2	0.216526	0.045624	0.037117	0.368804	0.097813	0.007730	0.458202	0.160674	0.332565	0.002960	0.006635	0.2	0.0	0.0	0.0	0.0	2	0.1	5
3	0.984937	0.008604	0.125205	0.330998	0.065548	0.393717	0.369659	0.055583	0.194877	0.003742	0.001138	0.2	0.0	0.0	0.0	0.0	3	0.1	5
4	0.661858	0.053289	0.408826	0.153988	0.063849	0.192904	0.463465	0.117601	0.316096	0.006090	0.021499	0.2	0.0	0.0	0.0	0.0	4	0.1	5

	SiO2_Liq	TiO2_Liq	Al2O3_Liq	FeOt_Liq	MnO_Liq	MgO_Liq	CaO_Liq	Na2O_Liq	K2O_Liq	Cr2O3_Liq	P2O5_Liq	H2O_Liq	Fe3Fet_Liq	NiO_Liq	CoO_Liq	CO2_Liq	Sample_ID_Liq
0	57.023602	0.623106	16.332899	4.36174	0.103851	4.19180	6.94858	3.59702	0.896895	0.000000	0.226584	5.59	0.0	0.0	0.0	0.0	0
1	57.658600	0.654150	17.194799	3.90621	0.084105	2.86892	5.91538	3.85948	1.018600	0.000000	0.214935	6.55	0.0	0.0	0.0	0.0	1
2	60.731201	0.862054	17.144199	4.07781	0.077488	2.50867	5.22075	4.45556	1.414160	0.000000	0.319638	3.14	0.0	0.0	0.0	0.0	2
3	61.532799	0.440860	16.508801	3.32990	0.037520	1.64150	4.34294	4.40860	1.407000	0.000000	0.215740	6.20	0.0	0.0	0.0	0.0	3
4	52.969101	0.803412	17.563000	5.93217	0.149472	3.78351	7.65110	3.80219	0.551178	0.037368	0.196182	6.58	0.0	0.0	0.0	0.0	4

Step 1 - Use function add_noise_sample_1phase() to add sample noise and make lots of synthetic liquids

• adds noise to myLiquids1 from a normal distribution with a mean defined by the measured value of each element, and the specified 1 sigma value (here, an absolute error given by the loaded in data myLiquids1_Err

• Specify number of resamples per liquid using “duplicates”. e.g., here, make 1000 synthetic liquids per measured liquid

• By default all negative numbers are replaced with zeros, but you can set Positive=False if you don’t want this behavoir

Liquids_only_abs_noise=pt.add_noise_sample_1phase(phase_comp=myLiquids1, phase_err=myLiquids1_Err,
                                             phase_err_type="Abs", duplicates=1000, err_dist="normal")

All negative numbers replaced with zeros. If you wish to keep these, set positive=False

• Here, we can look at all 1000 of the synthetic liquids generated for the first user-entered liquid (where Sample_Id=0)

Liquids_only_abs_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==0]

	SiO2_Liq	TiO2_Liq	Al2O3_Liq	FeOt_Liq	MnO_Liq	MgO_Liq	CaO_Liq	Na2O_Liq	K2O_Liq	Cr2O3_Liq	P2O5_Liq	H2O_Liq	Fe3Fet_Liq	NiO_Liq	CoO_Liq	CO2_Liq	Sample_ID_Liq_Num	Sample_ID_Liq
0	57.339119	0.570542	16.217459	4.344670	0.106114	4.214361	7.012736	3.132608	0.912690	0.004592	0.219173	5.654142	0.0	0.0	0.0	0.0	0.0	0
1	57.125681	0.539742	15.938553	4.301995	0.105458	4.346012	7.027890	3.324502	0.983291	0.000000	0.231536	5.351986	0.0	0.0	0.0	0.0	0.0	0
2	57.009565	0.559875	16.585746	4.369972	0.104239	3.632659	6.916967	3.961955	0.653859	0.002472	0.248753	5.410483	0.0	0.0	0.0	0.0	0.0	0
3	56.773927	0.717600	16.431267	4.365192	0.104798	4.294141	7.123197	3.684921	0.877529	0.015985	0.230147	5.898848	0.0	0.0	0.0	0.0	0.0	0
4	56.912873	0.667839	16.087009	4.422715	0.095112	4.067635	6.656492	3.612580	1.485076	0.000000	0.216596	5.379585	0.0	0.0	0.0	0.0	0.0	0
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
995	57.114978	0.732298	16.061787	4.212279	0.106915	3.812969	7.126286	2.087145	0.533055	0.013073	0.226516	5.632316	0.0	0.0	0.0	0.0	0.0	0
996	56.949382	0.684101	16.485557	4.327049	0.098417	3.968041	7.184717	3.736068	0.605825	0.001030	0.221936	5.497267	0.0	0.0	0.0	0.0	0.0	0
997	56.899779	0.701287	16.573631	3.974402	0.102925	3.537189	6.957506	4.039582	1.144930	0.000000	0.223950	5.405451	0.0	0.0	0.0	0.0	0.0	0
998	57.256831	0.746110	16.391035	4.240407	0.101621	4.936834	6.814363	3.575518	1.002793	0.000000	0.217243	5.664682	0.0	0.0	0.0	0.0	0.0	0
999	57.000129	0.513479	15.877145	4.173456	0.104173	4.761343	6.710175	3.246141	0.782031	0.008506	0.230655	5.572601	0.0	0.0	0.0	0.0	0.0	0

1000 rows × 18 columns

• We can plot some elements up against the user-entered 1 sigma to verify how the code is working

fig, ((ax1, ax2)) = plt.subplots(1, 2, figsize=(10, 4))

ax1.hist(Liquids_only_abs_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==0, 'MgO_Liq'],
         label='Synthetic', density=True)  ;
ax1.plot([myLiquids1['MgO_Liq'].iloc[0], myLiquids1['MgO_Liq'].iloc[0]], [0, 1], '-r', label='Meas')
ax1.plot([myLiquids1['MgO_Liq'].iloc[0]-myLiquids1_Err['MgO_Liq_Err'].iloc[0],
          myLiquids1['MgO_Liq'].iloc[0]-myLiquids1_Err['MgO_Liq_Err'].iloc[0]],
         [0, 1.5], ':r', label='Mean-1sigma entered')
ax1.plot([myLiquids1['MgO_Liq'].iloc[0]+myLiquids1_Err['MgO_Liq_Err'].iloc[0],
          myLiquids1['MgO_Liq'].iloc[0]+myLiquids1_Err['MgO_Liq_Err'].iloc[0]],
         [0, 1.5], ':r', label='Mean+1sigma entered')
ax1.set_xlabel('MgO Liquid')
ax1.legend()

ax2.hist(Liquids_only_abs_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==0, 'MnO_Liq'],
         label='Synthetic', density=True)  ;
ax2.plot([myLiquids1['MnO_Liq'].iloc[0], myLiquids1['MnO_Liq'].iloc[0]], [0, 110], '-r', label='Ent. Value')
ax2.plot([myLiquids1['MnO_Liq'].iloc[0]-myLiquids1_Err['MnO_Liq_Err'].iloc[0],
          myLiquids1['MnO_Liq'].iloc[0]-myLiquids1_Err['MnO_Liq_Err'].iloc[0]],
         [0, 110], ':r', label='Mean-1sigma entered')
ax2.plot([myLiquids1['MnO_Liq'].iloc[0]+myLiquids1_Err['MnO_Liq_Err'].iloc[0],
          myLiquids1['MnO_Liq'].iloc[0]+myLiquids1_Err['MnO_Liq_Err'].iloc[0]],
         [0, 110], ':r', label='Mean+1sigma entered')
ax2.set_xlabel('MnO Liquid')

Text(0.5, 0, 'MnO Liquid')

Step 2 - input this synthetic dataframe into the functions for calculating temperature

• Here, using equation 22 of Putirka (2008), where DMg ol-liq is calculated theoretically using DMg from Beattie (1993) so this can be used as an olivine-only thermometer

T_noise=pt.calculate_liq_only_temp(liq_comps=Liquids_only_abs_noise, equationT="T_Put2008_eq22_BeattDMg",
                                   P=5)

Step 3 - Plot the results

• In this plot we show the histogram for the temperature from each liquid for liquid 1, 2, 3, and 4 (for all 1000 synthetic liquids generated from each measured liquid)

• All synthetic liquids generated from a single input liquid have the same value of ‘Sample_ID_Liq_Num’

fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(15, 10))
ax1.annotate("Liquid 1", xy=(0.02, 0.9), xycoords="axes fraction", fontsize=14)
ax1.hist(T_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==0], bins=50, density=True);
ax2.annotate("Liquid 2", xy=(0.02, 0.9), xycoords="axes fraction", fontsize=14)
ax2.hist(T_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==1], bins=50, density=True);
ax3.annotate("Liquid 3", xy=(0.02, 0.9), xycoords="axes fraction", fontsize=14)
ax3.hist(T_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==2], bins=50, density=True);
ax4.annotate("Liquid 4", xy=(0.02, 0.9), xycoords="axes fraction", fontsize=14)
ax4.hist(T_noise.loc[Liquids_only_abs_noise['Sample_ID_Liq_Num']==3], bins=50, density=True);
ax1.set_xlabel('T (K)')
ax2.set_xlabel('T (K)')
ax4.set_xlabel('T (K)')
ax3.set_xlabel('T (K)')
ax1.set_ylabel('Probability Density')
ax3.set_ylabel('Probability Density')

Text(0, 0.5, 'Probability Density')

Step 4 - Use a function to get the mean, median and standard deviation for each liquid

The two arguements for this function are: 1) The panda series you want to average (in this case, temperature) 2) The panda series of values you want to average by, e.g., here averaging all samples with the same sample ID.

Stats_T_K=pt.av_noise_samples_series(T_noise, Liquids_only_abs_noise['Sample_ID_Liq_Num'])
Stats_T_K

	Sample	# averaged	Mean_calc	Median_calc	St_dev_calc	Max_calc	Min_calc
0	0.0	1000	1309.902145	1310.380062	13.093340	1346.365944	1268.209389
1	1.0	1000	1256.813446	1257.032875	17.370469	1304.352787	1193.437455
2	2.0	1000	1297.558476	1297.505347	5.320186	1313.744696	1280.892252
3	3.0	1000	1219.449607	1221.116468	23.046260	1278.109314	1119.734577
4	4.0	1000	1283.170342	1282.914869	7.092503	1307.488163	1258.829138
5	5.0	1000	1269.331119	1269.400970	3.307564	1279.595508	1258.864891
6	6.0	1000	1259.467489	1259.539682	5.811925	1278.471415	1241.752798
7	7.0	1000	1197.187020	1197.280186	4.130928	1210.318532	1182.177802
8	8.0	1000	1166.831116	1166.589258	6.831496	1188.374763	1146.182586
9	9.0	1000	1335.289418	1335.189830	9.228234	1368.818995	1300.345601

## You can plot these up too
plt.plot(Stats_T_K['Mean_calc'], Stats_T_K['St_dev_calc'], 'ok')
plt.xlabel('Mean T (K)')
plt.ylabel('1 sig error (K)')

Text(0, 0.5, '1 sig error (K)')

Example 2 - Making noise using percentage Errors

• Here, in the input spreadsheet we have specified a percentage error for each input (e.g., you could estimate this from EPMA analyses of secondary standards, or from EPMA software 1 sigma outputs).

out2=pt.import_excel('Liquid_Errors.xlsx', sheet_name="Error_Example_Perc")
my_input2=out2['my_input']
myOls2=out2['Ols']
myLiquids2=out2['Liqs']

out_Err2=pt.import_excel_errors('Liquid_Errors.xlsx', sheet_name="Error_Example_Perc")
myLiquids2_Err=out_Err2['Liqs_Err']
myinput2_Out=out_Err2['my_input_Err']

display(myLiquids2_Err.head())

	SiO2_Liq_Err	TiO2_Liq_Err	Al2O3_Liq_Err	FeOt_Liq_Err	MnO_Liq_Err	MgO_Liq_Err	CaO_Liq_Err	Na2O_Liq_Err	K2O_Liq_Err	Cr2O3_Liq_Err	P2O5_Liq_Err	H2O_Liq_Err	Fe3Fet_Liq_Err	NiO_Liq_Err	CoO_Liq_Err	CO2_Liq_Err	Sample_ID_Liq_Err	P_kbar_Err	T_K_Err
0	1	3	5	4	10	2	3	10	10	20	5	10	0.0	0.0	0.0	0.0	0	1	1
1	1	3	5	4	10	2	3	10	10	20	5	10	0.0	0.0	0.0	0.0	1	1	1
2	1	3	5	4	10	2	3	10	10	20	5	10	0.0	0.0	0.0	0.0	2	1	1
3	1	3	5	4	10	2	3	10	10	20	5	10	0.0	0.0	0.0	0.0	3	1	1
4	1	3	5	4	10	2	3	10	10	20	5	10	0.0	0.0	0.0	0.0	4	1	1

Step 1 - add errors based on the dataframe Liquid2_Err which are percentage errors.

• makes 1000 liquids per user-entered row, and assume errors are normally distributed

• The difference from example 1 is now we say phase_err_type=“Perc” not “Abs”

• Here, say Positive=False, which means it keeps negative numbers

Liquids_only_noise2=pt.add_noise_sample_1phase(phase_comp=myLiquids2, phase_err=myLiquids2_Err,
                     phase_err_type="Perc", duplicates=1000, err_dist="normal", positive=False)

# Validation that its calculating percentage errors right, as told it to add 1% error for SiO2.
#This will vary a bit as you run it, as its random
Mean=np.mean(Liquids_only_noise2.loc[Liquids_only_noise2['Sample_ID_Liq_Num']==0, 'SiO2_Liq'])
std_Dev=np.nanstd(Liquids_only_noise2.loc[Liquids_only_noise2['Sample_ID_Liq_Num']==0, 'SiO2_Liq'])
100*std_Dev/Mean

1.0320514822183176

Step 2 - calculating temperatures for all these synthetic liquids using equation 16 of Putirka (2008)

T_noise2=pt.calculate_liq_only_temp(liq_comps=Liquids_only_noise2, equationT="T_Put2008_eq16", P=5)

Step 3 - Calculating standard deviations, means, and medians etc

Stats_T_K2=pt.av_noise_samples_series(T_noise2, Liquids_only_noise2['Sample_ID_Liq_Num'])
Stats_T_K2

	Sample	# averaged	Mean_calc	Median_calc	St_dev_calc	Max_calc	Min_calc
0	0.0	1000	1361.549592	1360.828029	21.284278	1427.380228	1299.163312
1	1.0	1000	1283.911098	1282.884993	25.482757	1377.171448	1203.308920
2	2.0	1000	1290.465315	1289.814799	27.056968	1392.414592	1201.223386
3	3.0	1000	1264.118991	1264.126248	31.298034	1359.931213	1179.901757
4	4.0	1000	1328.369298	1327.090168	16.986712	1398.653276	1281.722698
5	5.0	1000	1310.432659	1310.717497	19.262560	1397.345604	1254.534783
6	6.0	1000	1292.872374	1292.289668	22.242435	1364.322919	1219.681620
7	7.0	1000	1377.059542	1375.925314	18.501446	1443.671512	1318.683633
8	8.0	1000	1280.766252	1280.072501	21.139515	1355.840834	1225.283955
9	9.0	1000	1380.484430	1380.051080	11.631417	1421.575492	1346.935368

Example 3: Fixed Percentage Error

• Here, instead of inputted columns with “_Err” for each oxide, we just add a fixed percentage error to all oxides (still normally distributed) - here, 1% noise for all elements

out3=pt.import_excel('Liquid_Errors.xlsx', sheet_name="Error_Example_Perc")
my_input3=out3['my_input']
myOls23=out3['Ols']
myLiquids3=out3['Liqs']

Liquids_only_noise3=pt.add_noise_sample_1phase(phase_comp=myLiquids3, duplicates=1000,
                           noise_percent=1, err_dist="normal")

All negative numbers replaced with zeros. If you wish to keep these, set positive=False

T_noise3=pt.calculate_liq_only_temp(liq_comps=Liquids_only_noise3, equationT="T_Put2008_eq16", P=5)

Stats_T_K3=pt.av_noise_samples_series(T_noise3, Liquids_only_noise3['Sample_ID_Liq_Num'])
Stats_T_K3

	Sample	# averaged	Mean_calc	Median_calc	St_dev_calc	Max_calc	Min_calc
0	0.0	1000	1361.134227	1361.014088	4.874867	1377.428997	1345.536425
1	1.0	1000	1283.517820	1283.640891	5.621473	1304.141379	1261.999314
2	2.0	1000	1289.716326	1289.416605	6.123934	1309.231428	1270.773626
3	3.0	1000	1262.756037	1263.181094	7.896493	1285.546263	1226.593845
4	4.0	1000	1327.388976	1327.359885	3.720417	1340.641267	1313.075183
5	5.0	1000	1310.250896	1310.103947	4.181113	1326.058457	1297.251874
6	6.0	1000	1292.005423	1291.920835	4.941247	1308.997040	1276.200752
7	7.0	1000	1375.892616	1375.747240	4.304133	1389.734398	1362.732758
8	8.0	1000	1279.485243	1279.484153	4.344912	1292.641127	1261.964871
9	9.0	1000	1381.030949	1380.944036	2.966598	1391.854457	1371.505604

Example 4: Perc uncertainty in a single input parameter

• Here, want to add 5% error to just MgO in the liquid. You don’t have to specify _Liq, it adds this based on what you entered.

Liquids_only_noise4=pt.add_noise_sample_1phase(phase_comp=myLiquids1, variable="MgO", variable_err=5,
                                              variable_err_type="Perc", duplicates=1000, err_dist="normal")

All negative numbers replaced with zeros. If you wish to keep these, set positive=False

T_noise4=pt.calculate_liq_only_temp(liq_comps=Liquids_only_noise4, equationT="T_Put2008_eq16", P=5)

Stats_T_K4=pt.av_noise_samples_series(T_noise4, Liquids_only_noise4['Sample_ID_Liq_Num'])
Stats_T_K4

	Sample	# averaged	Mean_calc	Median_calc	St_dev_calc	Max_calc	Min_calc
0	0.0	1000	1361.324037	1361.582552	6.392147	1378.325808	1335.150336
1	1.0	1000	1283.689998	1283.715019	5.468439	1301.527086	1266.072391
2	2.0	1000	1289.456819	1289.332550	4.431946	1305.173292	1273.517673
3	3.0	1000	1262.638115	1262.769202	3.150895	1272.859712	1252.423406
4	4.0	1000	1327.275815	1327.508849	6.504264	1346.903228	1304.674573
5	5.0	1000	1310.108091	1310.280959	5.949882	1333.104461	1291.936559
6	6.0	1000	1291.874784	1292.038464	5.050548	1309.241646	1276.252972
7	7.0	1000	1376.073899	1376.088693	7.466990	1398.559068	1356.357994
8	8.0	1000	1279.706683	1279.843350	6.927670	1302.585541	1259.975205
9	9.0	1000	1380.751258	1381.059421	8.486740	1409.936152	1350.552863

Example 5: Absolute uncertainty in a given input parameter

• Here, say uncertainty in H2O content of the liquid is +-1 wt%

Liquids_only_noise5=pt.add_noise_sample_1phase(phase_comp=myLiquids1, variable="H2O", variable_err=1,
                                              variable_err_type="Abs", duplicates=1000, err_dist="normal")

All negative numbers replaced with zeros. If you wish to keep these, set positive=False

plot a histogram of resulting H2O distribution

plt.hist(Liquids_only_noise5.loc[Liquids_only_noise4['Sample_ID_Liq_Num']==0, 'H2O_Liq'], bins=100, density= True);
plt.xlabel('H$_2$O Liq')
plt.ylabel('Probability Density')

Text(0, 0.5, 'Probability Density')

Now feed this into a thermometer

#Feed into a thermometer
T_noise5=pt.calculate_liq_only_temp(liq_comps=Liquids_only_noise5, equationT="T_Put2008_eq22_BeattDMg",
                              P=5)
plt.hist(T_noise5.loc[Liquids_only_noise5['Sample_ID_Liq_Num']==0], bins=100, density=True);
plt.ylabel('Probability Density')
plt.xlabel('T (K)')

Text(0.5, 0, 'T (K)')

Stats_T_K5=pt.av_noise_samples_series(T_noise5, Liquids_only_noise5['Sample_ID_Liq_Num'])
Stats_T_K5

	Sample	# averaged	Mean_calc	Median_calc	St_dev_calc	Max_calc	Min_calc
0	0.0	1000	1310.446098	1310.488990	14.985740	1354.822164	1259.449375
1	1.0	1000	1258.556146	1257.993042	13.132467	1309.475327	1218.984410
2	2.0	1000	1297.688214	1297.365790	14.523009	1345.233068	1254.768252
3	3.0	1000	1221.298201	1221.376857	12.551356	1262.408269	1185.549858
4	4.0	1000	1282.783981	1282.844986	13.828206	1329.398621	1245.518315
5	5.0	1000	1269.043852	1268.805220	14.267293	1312.629549	1228.703445
6	6.0	1000	1259.207648	1258.678080	13.493764	1312.263118	1216.051498
7	7.0	1000	1197.326114	1196.895067	12.169079	1233.714918	1161.138675
8	8.0	1000	1167.487727	1167.236479	11.414431	1201.618012	1134.335560
9	9.0	1000	1335.204744	1335.372128	15.733560	1399.270430	1283.412667

Example 6 - Uniformly distributed errors

• by default, the code assumes a normal distribution of errors, calculated using the user-inputted 1 sigma value

• you can also state err_dist=“uniform”, to generate uniformly distributed noise between +-inputted value

Liquids_only_noise6=pt.add_noise_sample_1phase(phase_comp=myLiquids1, variable="H2O", variable_err=0.5,
                                              variable_err_type="Abs", duplicates=1000, err_dist="uniform")

All negative numbers replaced with zeros. If you wish to keep these, set positive=False

plt.hist(Liquids_only_noise6.loc[Liquids_only_noise6['Sample_ID_Liq_Num']==0, 'H2O_Liq'], bins=100);
plt.ylabel('Probability Density')
plt.xlabel('T (K)')

Text(0.5, 0, 'T (K)')